Exact boundary controllability of 1D semilinear wave equations through a constructive approach

Abstract

The exact controllability of the semilinear wave equation $y_{tt}-y_{xx}+ f(y)=0$, $x\in (0,1)$ assuming that f is locally Lipschitz continuous and satisfies the growth condition $\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta$ for some $\beta$ small enough and $p=2$ has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with $p=3/2$, by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition $\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta$ for some $\beta$ small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

1 Introduction and main results

Let $\Omega :=(0,1)$ and let $T>0$. We set $Q_T:=\Omega \times (0,T)$. We consider the semilinear 1D wave equation

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt}-y_{xx}+ f(y) =0&\text {in } Q_T, \\&y(0,\cdot ) =0, \ \ y(1,\cdot ) =v&\text {in } (0,T), \\&(y(\cdot ,0), y_t(\cdot ,0))= (u_0,u_1)&\text {in } \Omega , \end{aligned} \right. \end{aligned}$$

(1)

where $(u_0, u_1) \in \varvec{V}:=H^1_0(\Omega ) \times L^2(\Omega )$ is a given initial state, $v\in H^1_0(0,T)$ is a control function and $f\in \mathcal {C}^0(\mathbb R)$ is a nonlinear function such that $|f(r)|\leqslant C(1+|r|)\ln ^2(2+|r|)$ for every $r\in {\mathbb {R}}$ and some $C>0$. Then, (1) admits a unique weak solution in $\mathcal {C}^0([0,T]; H^1(\Omega ))\times \mathcal {C}^1([0,T];L^2(\Omega ))$ (see [9]).

The system (1) is said to be exactly controllable at time $T>0$ if for any initial state $(u_0, u_1)\in \varvec{V}$ and target data $(z_0, z_1)\in \varvec{V}$, there exists a control function $v\in H^1_0(0,T)$ such that the associated solution to (1) satisfies $(y(\cdot ,T), y_t(\cdot ,T))=(z_0,z_1)$. The controllability time $T>0$ needs to be large enough in view of the finite speed of propagation of the solutions

Literature- The exact controllability for the linear wave equations is by now well-understood, see for instance the pioneer works by Russell [34], Lions [23, 24], Lagnese and Lions [25]; we also refer [5].

The first work concerning the controllability of finite dimensional nonlinear wave equations has been done by Markus [30] by the way of an implicit function theorem. Later on, this approach has been adapted for the local exact controllability of nonlinear wave equations by Chewning [10], Fattorini [19]. Global exact controllability for the semilinear wave equations in any space dimension has first been obtained by Zuazua [37,38,39] assuming that the nonlinear functions are globally Lipschitz and asymptotically linear, i.e., assuming that $\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /\vert r\vert <\infty$. For the boundary controllability case, this asymptotic assumption has been removed in [40] in the framework of the HUM method introduced by Lions coupled with a fixed point argument.

Theorem 1

[40, Theorem 2.1] Assume that $T>2$. Then, for every globally Lipschitz continuous function f such that $f^\prime \in L^{\infty }(\mathbb R)$ and $\gamma \in (0,1)$, $\gamma \ne \frac{1}{2}$, the system (1) is exactly controllable in $H^{\gamma }_0(0,1) \times H^{\gamma -1}(0,1)$ with a control $v\in H^{\gamma }_0(0,T)$.

Later on, this result (actually proved in a multidimensional situation) was recovered by I. Lasiecka and R. Triggiani, [28], using a global inversion theorem. The authors improved some regularity of their boundary control still assuming globally Lipschitz nonlinearity.

Then, in the framework of the distributed controllability with a control support $\omega \subset (0,1)$, the assumption $f^\prime \in L^\infty (\mathbb {R})$ has been relaxed by Zuazua.

Theorem 2

[41, Theorem 1] Let $\omega =(l_1,l_2)$ with $0<l_1<l_2<1$. Assume that $T>2\max \{l_1, 1-l_2\}$, that f is locally Lipschitz continuous and satisfies

$(\mathbf{H}_\mathbf{1})$:

$\limsup _{\vert r\vert \rightarrow \infty } \frac{\vert f(r)\vert }{\vert r\vert \ln _+^{2}\vert r\vert }\leqslant \beta$

for some $\beta$ small enough depending only on $\omega$ and T. Then, for any $(u_0,u_1), (z_0,z_1)\in \varvec{V}$, the system

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt}-y_{xx}+ f(y) =v\, 1_{\omega }&\text {in } Q_T, \\&y(0,\cdot )=y(1,\cdot )=0&\text {in } (0,T), \\&(y(0,\cdot ), y_t(0,\cdot ))= (u_0,u_1)&\text {in } \Omega , \end{aligned} \right. \end{aligned}$$

(2)

is exactly controllable with control in $L^2(\omega \times (0,T))$: there exists $v\in L^2(\omega \times (0,T))$ such that $(y(\cdot ,T),y_t(\cdot ,T))=(z_0,z_1)$.

Here and in the sequel, we note

$$\begin{aligned} \ln _+|r|=\left\{ \begin{aligned}&0 \hbox { if } |r|\leqslant 1\\&\ln |r| \hbox { else}. \end{aligned}\right. \end{aligned}$$

Moreover, it is proved in [41, Theorem 2] that, if f behaves like $-s\ln ^p(\vert s\vert )$ with $p>2$ as $\vert s\vert \rightarrow +\infty$, then the system is not exactly controllable in any time $T>0$, due to an uncontrollable blow-up phenomenon. Theorem 1 has been slightly improved in [12], weakening the condition $(\mathbf{H}_\mathbf{1})$ into

$$\begin{aligned} \limsup _{\vert r\vert \rightarrow +\infty } \ \left| \int _0^r f(r^\prime )\, dr^\prime \right| \ \left( \vert r\vert \prod _{k=1}^{+\infty } \ln ^{[k]}(e_k+r^2)\right) ^{-2}<+\infty \end{aligned}$$

where $\ln ^{[k]}$ denotes the $k^{\text {th}}$ iterate of $\ln$ and $e_k>0$ is such that $\ln ^{[k]}(e_k)=1$. This growth condition is optimal since the solution of (2) may blow up whenever f grows faster at infinity and has the bad sign. The multi-dimensional case in which $\Omega$ is a bounded domain of $\mathbb {R}^d$, $d>1$, with a $\mathcal {C}^{1,1}$ boundary has been addressed in [29]; assuming that the support $\omega$ of the control function is a neighborhood of $\partial \Omega$ and that $T>\text {diam}(\Omega \backslash \omega )$, the exact controllability of (2) is proved under the growth condition $\limsup _{\vert r\vert \rightarrow +\infty } \frac{\vert f(r)\vert }{\vert r\vert \ln _+^{1/2}\vert r\vert } < \beta$ for some $\beta$ small enough. For control domains $\omega$ satisfying the classical multiplier assumption (see [24]), exact controllability has been proved in [36] assuming that f is globally Lipschitz continuous. We also mention [14] where a positive boundary controllability result is proved for steady-state initial and final data and for T large enough by a quasi-static deformation approach. We also mention the work by Dehman et al. [16] which is concerned with the controllability and stabilizability of some subcritical semilinear wave equations in $\Omega ^\prime \subset \mathbb R^3$. Assuming that the nonlinearity $f\in \mathcal {C}^3(\mathbb R)$ satisfies

$$\begin{aligned} f(0) =0, \ \ r f(r) \geqslant 0, \ \ |f^{(j)}(r)| \leqslant C(1+|r|)^{p-j}, \ \ j=1,2,3; \ \ 1\leqslant p< 5 \end{aligned}$$

the exact internal controllability of the semilinear wave equations at time $T:=T(u_0,u_1)>0$ that depends on the size of the initial data $(u_0,u_1)\in H^1_0(\Omega ^\prime ) \times L^2(\Omega ^\prime )$. See also [15] achieving the same result in a uniform time under smallness assumption on the initial data. The sign condition has been weakened in [21] to an asymptotic sign assumption leading to a semi-global internal controllability result in the sense that the target data are prescribed in a precise subset of $H^1_0(\Omega ^\prime )\times L^2(\Omega ^\prime )$ .

The above results devoted to internal controllability, notably Theorem 2, can be employed together with the domain extension method to get indirectly boundary controllability results for system (1) of interest in the present work.

The proof of Theorem 2 is based on a Leray Schauder fixed point argument applies to the operator $\Lambda :L^\infty (Q_T)\rightarrow L^\infty (Q_T)$, where $y:=\Lambda (z)$ is a controlled solution with the control function v of the linear boundary value problem

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt}- y_{xx}+ \widehat{f}(z)\,y= -f(0)+v 1_{\omega }&\text {in}\ Q_T,\\&y(0,\cdot )=y(1,\cdot )=0&\text {in}\ (0,T), \\&(y(\cdot ,0),y_t(\cdot ,0))=(u_0,u_1)&\text {in}\ \Omega , \end{aligned} \right. \qquad \widehat{f}(r):= \left\{ \begin{aligned}&\frac{f(r)-f(0)}{r}&\text {if}\ r\ne 0\\&f^{\prime }(0)&\text {if}\ r=0 \end{aligned} \right. \nonumber \\ \end{aligned}$$

(3)

satisfying $(y(\cdot ,T),y_t(\cdot ,T))=(z_0,z_1)$. The stability of the operator $\Lambda$ in $L^\infty (Q_T)$ is based on a precise estimate of the cost of the control v in terms of the potential $\widehat{f}$ and data $(u_0,u_1)$, $(z_0,z_1)$.

Objective - The general goal addressed in this work is the approximation of the controllability problem associated with (1), that is to construct an explicit sequence $(y_k,v_k)_{k\in \mathbb {N}}$ converging strongly toward a control-state pair solution (y, v) for (1). Although almost sharp with respect to the nonlinearity, Theorem 2 is not constructive as it does not provide any convergent sequences $(y_k)_{k\in \mathbb {N}}$ to a fixed point of $\Lambda$, i.e., to a controlled solution y of (2). This is due to the fact that the operator $\Lambda$ is not contracting in general.

Assuming slightly stronger assumptions on f, a constructive convergent sequence has been proposed by the third author and E. Trélat in [32] using a least-squares approach coupled with a Newton type linearization.

Theorem 3

[32, Theorem 2.3] Let $\omega =(l_1,l_2)$ with $0<l_1<l_2<1$. Assume that $T>2\max \{l_1, 1-l_2\}$ and that $f\in \mathcal {C}^{1}(\mathbb R)$ satisfies

$(\mathbf{H}_\mathbf{1})^{\prime }$:

$\exists \alpha >0, \text { s.t. } \vert f^\prime (r)\vert \leqslant \alpha +\beta \ln ^2_{+}\vert r\vert , \quad \forall r\in \mathbb {R}$

for some $\beta >0$ small enough depending only on $\omega$ and T and

$(\overline{\mathbf{H}}_\mathbf{p})$:

$\displaystyle \exists p\in (0,1] \text { such that } \sup _{\begin{array}{c} a,b\in \mathbb {R}\\ a\ne b \end{array}} \frac{\vert f^\prime (a)-f^\prime (b)\vert }{\vert a-b\vert ^p} < +\infty$.

Then, for any $(u_0,u_1), (z_0,z_1) \in \varvec{V}$, one can construct a sequence $(y_k,v_k)_{k\in \mathbb {N}}$ converging strongly to a controlled pair for (2) satisfying $(y(\cdot ,T),y_t(\cdot ,T))=(z_0,z_1)$. Moreover, after a finite number of iterations, the convergence is of order at least $1+p$.

The hypothesis on f is stronger here than in Theorem 1: it should be noted, however, that the function $f(r)=a+br+ \beta r\ln (1+\vert r\vert )^2$, $a,b\in \mathbb {R}$ which is somehow the limit case in $(\mathbf{H}_\mathbf{1})$ satisfies $(\mathbf{H}_\mathbf{1})^{\prime }$ and $\mathbf (\overline{H}_1)$. On the other hand, Theorem 3 is constructive, contrary to Theorem 1. The construction makes appear the operator $\Lambda _1:L^\infty (Q_T)\rightarrow L^\infty (Q_T)$ where $y:=\Lambda _1(z)$ is a controlled solution with the control function v of the linear boundary value problem

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt}-y_{xx}+ f^{\prime }(z)y=v1_{\omega } + f^{\prime }(z)z-f(z)&\text {in}\ Q_T,\\&y(0,\cdot )=y(1,\cdot )=0&\text {in}\ (0,T), \\&(y(\cdot ,0),y_t(\cdot ,0))=(u_0,u_1)&\text {in}\ \Omega . \end{aligned} \right. \end{aligned}$$

(4)

Theorem 3 is extended to a multidimensional case, i.e., $\Omega \subset \mathbb {R}^d$ with $d\leqslant 3$ in [4] under the same condition on f except that the exponent 2 in $(\mathbf{H}_\mathbf{1})^{\prime }$ is replaced by an exponent 1/2.

Main result of the present work - In this paper, we prove the following result, directly in the framework of the boundary controllability.

Theorem 4

Assume $T>2$. Let $s>0$ large enough.

• There exists $\beta ^\star >0$ such that if $f \in \mathcal {C}^0(\mathbb {R})$ satisfies

$(\mathbf{H}_\mathbf{2})$:

$\exists \alpha _1,\alpha _2>0, \text { s.t. } \vert f(r)\vert \leqslant \alpha _1 + |r|\left( \alpha _2 + \beta ^\star \ln _+^{3/2}|r|\right) , \quad \forall r \in \mathbb R$

then system (1) is exactly controllable at time T for initial data in $\varvec{V}$ with controls in $H_0^1(0,T)$.

• There exists $\beta ^\star >0$ such that if f is locally Lipschitz and satisfies

$(\mathbf{H}_\mathbf{2}^\prime )$:

$\exists \alpha >0, \text { s.t. } \vert f^\prime (r)\vert \leqslant \alpha + \beta ^\star \ln _+^{3/2}|r| , \quad \forall r \in \mathbb R$

then, for any initial state $(u_0, u_1)$ and final state $(z_0,z_1)$ in $\varvec{V}$, one can construct a sequence $(y_k, v_k)_{k\in \mathbb N^*}$ that converges strongly to a controlled pair (y, v) in $\big (\mathcal {C}^0([0,T];H^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))\big )\times H_0^1(0,T)$ for the system (1). Moreover, the convergence of $(y_k,v_k)$ holds at least with a linear rate for the norm $\Vert \rho (s)\cdot \Vert _{L^2(Q_T)}+\Vert \rho _1(s)\cdot \Vert _{L^2(0,T)}$ where $\rho ,\rho _1$ are defined in (8) and s is chosen sufficiently large depending on $\Vert (u_0,u_1)\Vert _{\varvec{V}}$ and $\Vert (z_0,z_1)\Vert _{\varvec{V}}$.

To our knowledge, this result is the first proposing a constructive approximation of boundary controls for the semilinear wave equation without the assumption that f is globally Lipschitz. Under smallness assumptions on the data, we mention the recent works [7] and [33]. As in Theorem 2, the parameter $\beta ^\star$ have to be small enough depending only $\Omega$ and T through the constant appearing in a Carleman estimates (we refer to Remark 5). Moreover, the lower bound of the Carleman parameter s depends logarithmically on the size $\Vert (u_0,u_1)\Vert _{\varvec{V}}$ and $\Vert (z_0, z_1)\Vert _{\varvec{V}}$ of the data (we refer to Remark 6).

Concerning the first part of the theorem, if we compare with Theorem 2 (leading indirectly to boundary controllability result by the extension method), the assumption on the asymptotic behavior of f is slightly stronger with an exponent $\frac{3}{2}$ instead of 2. This is due to the fact the method in [41] based on explicit computation (using the d’Alembert formula) is genuinely one dimensional, while the present work is based on Carleman estimates valid in any space dimension. On the other hand, this first part relaxed the regularity assumption to $f\in \mathcal {C}^0(\mathbb {R})$ instead of f locally Lipschitz continuous. Moreover, this first part of Theorem 4 differs from Theorem 2 on the functional spaces as it is based on a different fixed point application leading to a simpler proof. In particular, it is not based on the analysis of the cost of observability of the wave equation with potential. Concerning the second part of the theorem, it relaxes the Hölder assumption $\mathbf (\overline{H}_p)$ on $f^\prime$ appearing in Theorem 3 but still leads to a constructive method. As we shall see, this is related to an appropriate choice of the parameter s related to the norm of the initial condition. Again, to our knowledge, this is the first result leading to a convergent approximation of boundary controls for superlinear nonlinearities without smallness assumption notably on the initial condition and target (contrary to the recent works [7] and [33]).

Theorem 4 is obtained by adapting the recent work [17] devoted to a semilinear heat equation. We introduce the following linearized controllability problem: for ${\widehat{y}}$ in a suitable class $\mathcal {C}_R(s)$ depending on a free parameter $s \geqslant 1$, find the control v such that the solution y of

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt}- y_{xx}= - f({\widehat{y}})&\text {in }Q_T,\\&y(0,\cdot )=0, \ y(1,\cdot )=v&\text {in } (0,T),\\&(y(\cdot ,0),y_t(\cdot ,0))=(u_0,u_1)&\text {in } \Omega , \end{aligned} \right. \end{aligned}$$

(5)

satisfies $(y(\cdot , T),y_t(\cdot ,T)) = (z_0,z_1)$ in $\Omega$, and (y, v) corresponds to the minimizer of a functional $J_s$ depending on s and involving Carleman weight functions (see Remark 3).

This will define an operator $\Lambda _s: {\widehat{y}} \mapsto y$ from some suitable class $\mathcal {C}_R(s)$ into itself, on which we can use fixed point theorems for s sufficiently large depending on $\Vert (u_0,u_1)\Vert _{\varvec{V}}$, namely Schauder fixed point theorem for the first item of Theorem 4, and Banach-Picard fixed point theorem for the second item, allowing to exhibit a simple sequence of convergent approximations of the control and controlled trajectory. The analysis of the fixed point operator is based on Carleman estimates as they allow to get precise estimates on the control and controlled trajectories in terms of the parameter s. Choosing the Carleman parameter large allows to limit the influence of lowers order terms and get suitable contracting properties. Such tricks have already been used in the context of inverse problems reformulated through a least-squares functional in [1] and [22].

With respect to the heat equation considered in [17, 27], the Carleman weights are not singular with respect to the time variable, avoiding technicalities. On the other hand, the regularity issue is more delicate for the hyperbolic case which does not enjoys regularizing property. This is a fortiori true for boundary control : precisely, in order to get $L^\infty$ estimate for the controlled trajectories solution of (1), the boundary control v needs to be more regular than $L^2(0,T)$. Hopefully, it turns out that the optimal state-control pair for the functional $J_s$ (see Remark 3) involving $L^2$ norms enjoys suitable regularity property as soon as the initial and final data belongs to $\varvec{V}$ and satisfy compatibility condition at $x=0$ and $x=1$. This point is crucial in our analysis.

Outline- The paper is organized as follows. In Sect. 2, we derive a controllability result for the linear wave equation with precise estimates in term of the right hand side, the initial data and the Carleman parameter s large enough. In particular, we prove that the optimal control for the $L^2(0,T)$ norm belongs actually to $H^1(0,T)$: this result stated in Theorem 7 is proven in Appendix A. Then, in Sect. 3, we prove, for any time $T>2$ the uniform null controllability of (1) assuming that f is continuous and satisfies the condition $(\mathbf{H}_\mathbf{2})$. Then, in Sect. 3.5, assuming that $f^\prime$ is continuous and satisfies the condition $(\mathbf{H}_\mathbf{1})^{\prime }$, we show that the operator $\Lambda _s$ is contracting, yielding the convergence of the Picard iterates $y_{k+1}=\Lambda _s(y_k)$. Section 5 illustrates the result with some numerical experiments, while Sect. 6 concludes with some remarks.

Notations- In this article, C denote generic constants depending on $\Omega$ and T, which may change from line to line, but are independent of the Carleman parameter s.

2 Controllability results for the linear wave equation

This section is devoted to a controllability result for a linear wave equation with a right hand side $B\in L^2(Q_T)$ and initial data $(u_0,u_1)\in \varvec{V}$. Precisely, for any $(z_0,z_1)\in \varvec{V}$ and $T>0$ large enough, we are interested by the existence of a control function $v\in H_0^1(0,T)$ such that the solution y of

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt}-y_{xx}=B&\text {in } Q_T, \\&y(0,\cdot )=0, \ y(1,\cdot )=v&\text {in}\ (0,T), \\&(y(\cdot ,0),y_t(\cdot ,0)) = (u_0,u_1)&\text {in } \Omega , \end{aligned} \right. \end{aligned}$$

(6)

satisfies $(y(\cdot ,T), y_t(\cdot ,T))=(z_0,z_1)$. Though this linear control is by now standard, we aim to get precise weighted estimate of a state-control pair in a given functional space in the data, which will be crucial to handle the nonlinear system (1). We employ Carleman estimates as fundamental tool (see [6]).

2.1 A global Carleman estimate

For any $\beta \in (0,1)$ and $x_0<0$, we define the auxiliary function

$$\begin{aligned} \psi (x,t) = |x-x_0|^2 -\beta \biggl (t-\frac{T}{2}\biggr )^2 + M_0 \quad \text {in } Q_T, \end{aligned}$$

(7)

where $M_0>0$ is chosen in such a way that $\psi$ is strictly positive. Then, for any $\lambda >0$, we define $\phi (x,t)=e^{\lambda \psi (x,t)}$. For all $s\geqslant s_0$, let us now define the following weight functions

$$\begin{aligned} \rho (s;x,t):= e^{-s\phi (x,t)}, \quad \rho _1(s;t)=\rho (s;1,t), \quad \forall (x,t) \in Q_T. \end{aligned}$$

(8)

Remark that

$$\begin{aligned} e^{-cs}\leqslant \rho (s;x,t)\leqslant e^{-s}, \quad e^{-cs}\leqslant \rho _1(s;t)\leqslant e^{-s} \quad \text {in } Q_T \end{aligned}$$

with $c:=\Vert \phi \Vert _{L^\infty (Q_T)}$, that $\rho ^{-1},\rho \in \mathcal {C}^\infty (\overline{Q_T})$ and that $\rho _1,\rho _1^{-1}\in \mathcal {C}^\infty ([0,T])$. In short, we shall write $\rho (s)$ and $\rho _1(s)$ to denote the above weight functions.

Then, for any $\delta >0$ such that $T- 2\delta > 2 \sup _{\overline{\Omega }}|x-x_0|$, we introduce a cut-off function $\eta \in \mathcal {C}^\infty _c(\mathbb R)$ satisfying the following properties:

(9)

Let

$$\begin{aligned} P:=\Big \{w\in \mathcal {C}^0([0,T];H_0^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega )), w_{tt}-w_{xx}\in L^2(Q_T)\Big \}. \end{aligned}$$

Recall that $w_x(1,\cdot )\in L^2(0,T)$ for every $w\in P$ (see [23, Theorem 4.1]).

The controllability property for the linear system (6) is based on the following Carleman estimate with boundary observation at $x=1$.

Theorem 5

Assume $T>2$. There exists $s_0>0$, $\lambda >0$ and $C>0$, such that for any $s\geqslant s_0$, we have the following Carleman inequality

$$\begin{aligned}&s \int _{Q_T}\rho ^{-2}(s) (| w_{t}|^2 + | w_x|^2) + s^3 \int _{Q_T}\rho ^{-2}(s) | w|^2 \nonumber \\&\qquad + s \int _\Omega \rho ^{-2}(s;\cdot ,0) (|w_t(\cdot ,0)|^2 + |w_x(\cdot ,0)|^2) + s^3 \int _\Omega \rho ^{-2}(s;\cdot ,0) |w(\cdot ,0)|^2 \nonumber \\&\qquad + s \int _\Omega \rho ^{-2}(s;\cdot ,T) (|w_t(\cdot ,T)|^2 + |w_x(\cdot ,T)|^2) \,\text {d}x+ s^3 \int _\Omega \rho ^{-2}(s;\cdot ,T) |w(\cdot ,T)|^2 \nonumber \\&\quad \leqslant C \int _{Q_T}\rho ^{-2}(s) |w_{tt}-w_{xx}|^2 + C s \int _0^T \eta ^2(t)\rho ^{-2}_1(s) |w_x(1,\cdot )|^2 \end{aligned}$$

(10)

for every $w\in P$.

Proof

We refer to [8, Lemma 2.3] using [1, Remark 2.9 and Theorem 2.5]. Remark that the occurrence of the terms at $t=T$ on the left hand side are due to the fact that $\rho (\cdot ,t)=\rho (\cdot ,T-t)$ and the reversibility of the wave operator. $\square$

2.2 Application to controllability

In a standard way, Theorem 5 allows to deduce some controllability results for the system (6). For any $s\geqslant s_0$, we define the bilinear form

$$\begin{aligned} (w,z)_{P,s} := \int _{Q_T}\rho ^{-2}(s) Lw Lz + s \int _0^T \eta ^2(t)\rho _1^{-2}(s) w_x(1,t) z_x(1,t), \end{aligned}$$

(11)

for any $w,z\in P$. Here and in what follows, we use the notation $Lw:=w_{tt}-w_{xx}$. It is easily seen that (11) defines a scalar product in P and if $P_s$ denotes P endowed with this scalar product, then $P_s$ is an Hilbert space.

We can state the main result of this section, devoted without loss of generality to the null controllability case, for which $(z_0,z_1)=(0,0)$ in $\Omega$.

Theorem 6

Assume that $T>2$ and let $\eta \in \mathcal {C}^\infty (\mathbb {R})$ be a cut-off function satisfying (9). For $s\geqslant s_0$, $B \in L^2(Q_T)$ and $(u_0, u_1) \in \varvec{V}$, there exists unique $w_s \in P_s$, depending only on $B, u_0, u_1$ such that

$$\begin{aligned} (w_s, z)_{P,s}= \int _{\Omega } u_1\,z(\cdot ,0)\,\text {d}x- \int _\Omega u_0\, z_t(\cdot ,0)\,\text {d}x+ \int _{Q_T}B z, \quad \forall z\in P_s. \end{aligned}$$

(12)

Then, $v_s(t) = s\eta ^2(t)\rho _1^{-2}(s)(w_s)_x(1,t)$ is a control function for (6) where $y_s=\rho ^{-2}(s)L w_s$ is the associated controlled trajectory, that is $y_s(x,T)=(y_s)_t(x,T)=0$ for all $x\in \Omega$ and the operator defined by

$$\begin{aligned} \Lambda ^0_s: (B, u_0, u_1) \mapsto y_s \end{aligned}$$

(13)

is linear, continuous from $L^2(Q_T) \times H^1_0(\Omega ) \times L^2(\Omega )$ to $L^2(Q_T)$.

Moreover, we have the following estimates for $y_s$ and $v_s$ for some constant $C>0$ independent of s:

$$\begin{aligned} \begin{aligned} \Vert \rho (s) y_s\Vert _{L^2(Q_T)}&+ s^{-1/2}\left\| \frac{\rho _1(s)}{\eta } v_s\right\| _{L^2(\delta ,T-\delta )}\\&\leqslant C \Bigg (s^{-3/2} \Vert \rho (s)B\Vert _{L^2(Q_T)} + s^{-1/2} e^{-s} \Vert u_0\Vert _{L^2(\Omega )} + s^{-3/2} e^{-s} \Vert u_1\Vert _{L^{2}(\Omega )} \Bigg ). \end{aligned}\nonumber \\ \end{aligned}$$

(14)

Before going to the proof, we make the following remarks.

Remark 1

It is well-known that the boundary controllability of (6) with $L^2(0,T)$ controls holds true with initial data $(u_0,u_1)$ only in $L^2(\Omega )\times H^{-1}(\Omega )$. We start directly with $(u_0,u_1)\in \varvec{V}$ since the application of some fixed point theorem to deal with the semilinear case shall require regularity on the state-control pair (see Sect. 2.3).

Remark 2

In the framework of exact controllability with no vanishing target $(z_0,z_1)\in \varvec{V}$, the right hand side of estimate (14) contains the extra quantities

$$\begin{aligned} s^{-1/2} e^{-s} \Vert z_0\Vert _{L^2(\Omega )} + s^{-3/2} e^{-s} \Vert z_1\Vert _{L^2(\Omega )}. \end{aligned}$$

The point here to be noted is that the coefficients (powers of s or exponentials associated with s) in front of the norms of $u_0$, $u_1$ and $z_0$, $z_1$ are the same, and this would hold for any subsequent estimates.

This is why, there is no loss of generality to choose $(z_0,z_1)=(0,0)$ which will make the computations shorter and simpler.

Proof

We first ensure the solvability of the variational Eq. (12). Since $(\cdot , \cdot )_{P,s}$ is a scalar product on $P_s$, we only need to check that the right hand side of (12) is a linear continuous form on $P_s$.

• For all $z\in P_s$: since $\rho (s)B \in L^2(Q_T)$, we have

  $$\begin{aligned} \left| \int _{Q_T}B z \right| \leqslant \left( \int _{Q_T}|\rho (s) B|^2 \right) ^{1/2} \left( \int _{Q_T}|\rho ^{-1}(s) z|^2 \right) ^{1/2} . \end{aligned}$$

  Now, since z enjoys the Carleman inequality (10), one has $\Vert \rho ^{-1}(s)z\Vert _{L^2(Q_T)} \leqslant C s^{-3/2} \Vert z\Vert _{P_s}$ (recall the definition of the inner product (11) on $P_s$). Thus, we have

  $$\begin{aligned} \left| \int _{Q_T}B z \right| \leqslant C s^{-3/2} \Vert \rho (s)B\Vert _{L^2(Q_T)} \Vert z\Vert _{P_s}. \end{aligned}$$

• Next, we observe that

  $$\begin{aligned} \left| \int _\Omega u_0 z_t(\cdot ,0) \,\text {d}x\right|&\leqslant \Vert \rho (s;\cdot ,0) u_0\Vert _{L^2(\Omega )} \Vert \rho ^{-1}(s;\cdot ,0) z_t(\cdot ,0)\Vert _{L^2(\Omega )} \\&\leqslant C s^{-1/2} e^{-s} \Vert u_0\Vert _{L^2(\Omega )}\Vert z\Vert _{P_s}, \end{aligned}$$

  using the Carleman inequality (10) and that $|\rho (s;x,0)|=|e^{-s\phi (x,0)}|\leqslant e^{-s}$ (since $\phi \geqslant 1$).

• Similarly, we get

  $$\begin{aligned} \left| \int _\Omega u_1 z(\cdot ,0) \,\text {d}x\right|&\leqslant \Vert \rho (s;\cdot ,0) u_1\Vert _{L^2(\Omega )} \Vert \rho ^{-1}(s;\cdot ,0) z(\cdot ,0)\Vert _{L^2(\Omega )} \\&\leqslant C s^{-3/2} e^{-s} \Vert u_1\Vert _{L^2(\Omega )} \Vert z\Vert _{P_s}. \end{aligned}$$

Combining the above three items, the right hand side of (12) corresponds to a linear functional on $P_s$. The Riesz representation theorem implies the existence of a unique $w_s\in P_s$ satisfying the formulation (12) which additionally satisfies

$$\begin{aligned} \Vert w_s\Vert _{P_s} \leqslant C \left( s^{-3/2} \Vert \rho (s)B\Vert _{L^2(Q_T)} + s^{-1/2} e^{-s} \Vert u_0\Vert _{L^2(\Omega )} + s^{-3/2} e^{-s} \Vert u_1\Vert _{L^2(\Omega )} \right) , \end{aligned}$$

(15)

where the constant $C>0$ is independent of $s\geqslant s_0$.

Then, set $y_s= \rho ^{-2}(s)Lw_s$ and $v_s= s\eta ^2\rho ^{-2}(s)(w_s)_{x}(1,\cdot )$. From the equality (12), the pair $(y_s, v_s)$ satisfies

$$\begin{aligned} \int _{Q_T}y_s L z \,\text {d}x\text {d}t+ \int _0^T v_s z_x(1,\cdot ) \text {d}t&= \int _\Omega u_1\, z(\cdot ,0) \,\text {d}x- \int _\Omega u_0\, z_t(\cdot ,0) \,\text {d}x\\ {}&\quad + \int _{Q_T}B z, \ \ \forall z \in P_s, \end{aligned}$$

meaning that $y_s \in L^2(Q_T)$ is a solution to the linear system (6) associated with the function $v_s\in L^2(0,T)$ in the sense of transposition. By uniqueness, $y_s$ indeed solves (6) in a weak sense. Eventually, using the estimate (15) for $w_s$, we get that $\rho (s)y_s=\rho ^{-1}(s) Lw_s \in L^2(Q_T)$ and $s^{-1/2}\rho _1(s) v_s= s^{1/2} \eta ^2\rho _1^{-1}(s) (w_s)_x(1,\cdot ) \in L^2(0,T)$ and deduce the weighted estimate (14). $\square$

Remark 3

The functions $y_s$ and $v_s$ introduced by Theorem 6 can be characterized as the unique minimizer of the following functional

$$\begin{aligned} J_s(y,v) := s\int _{Q_T}\rho ^2(s) y^2 + \int _{\delta }^{T-\delta } \eta ^{-2} \rho ^{2}_1(s) v^2 \end{aligned}$$

(16)

over the set $\Big \{ (y,v): y \!\in \! L^2(Q_T), \eta ^{-1}v\!\in \! L^2(\delta , T-\delta ) \text { solution of } (6) \text { with } y(\cdot ,T)= y_t(\cdot ,T)=0 \text { in }\Omega \Big \}$. We refer to [8, Section 2] for the details.

Remark 4

The controlled state $y_s= \rho ^{-2} L w_s$ satisfies

(17)

implying (by standard regularity results for the wave equation) that $y_s\in \mathcal {C}^0([0,T];L^2(\Omega )) \cap \mathcal {C}^1([0,T];H^{-1}(\Omega ))$. On the other hand, the function $w_s$ uniquely satisfies the equation

(18)

implying that $(w_s(\cdot ,0),\partial _t w_s(\cdot ,0))\in \varvec{V}$ (see estimate (10)) and $w_s \in \mathcal {C}^0([0,T]; H^1_0(\Omega )) \cap \mathcal {C}^1([0,T];L^{2}(\Omega ))$.

2.3 Estimate for the state-control pair in $\mathcal {C}^0([0,T];H^1(\Omega ))\times H^1(0,T)$

In this section, we prove that the state-control pair $(y_s,v_s)$ given by Theorem 6 enjoys additional regularity property, under the assumption $(u_0, u_1) \in \varvec{V}$ and the introduction of the cut-off function $\eta$ with respect to the time variable. In particular, we obtain that $v_s\in H_0^1(0,T)$ and $y_s\in L^\infty (Q_T)$. This gain is crucial for the analysis of the semilinear case.

Theorem 7

Let any $(u_0, u_1)\in \varvec{V}$ and $B \in L^2(Q_T)$ be given. Then, the solution $(y_s,v_s)$ of (6) defined in Theorem 6 satisfies $v_s\in H^1(0,T)$, $y_s\in \mathcal {C}^0([0,T];H^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$ and the following estimate :

$$\begin{aligned} \begin{aligned}&\Vert \rho (s) (y_s)_t \Vert _{L^2(Q_T)} + s^{-1/2} \Vert \rho _1(s) (v_s)_t \Vert _{L^2(0,T)}\\&\quad \leqslant C \left( s^{-1/2} \Vert \rho (s)B\Vert _{L^2(Q_T)} + s^{-1/2} e^{-s} \Vert u_1\Vert _{L^2(\Omega )} \right. \\&\qquad \left. +s^{1/2}e^{-s} \Vert u_0\Vert _{L^2(\Omega )} + s^{-1/2} e^{-s} \Vert (u_0)_x\Vert _{L^2(\Omega )}\right) . \end{aligned} \end{aligned}$$

(19)

We refer to [18, Theorem 5.4] where a similar gain of regularity is proved in the simpler case of control of minimal $L^2(0,T)$-norm, i.e., $J_s$ in (16) is replaced by $J(y,v)=\Vert v\Vert ^2_{L^2(0,T)}$. We also refer to [15] for internal control by introducing a cut-off function in space. The proof of Theorem 7 is long and requires several steps. It is done in Appendix A.

Let us prescribe the following regularity estimate for the controlled trajectory $y_s$.

Lemma 1

Let us recall the controlled trajectory $y_s$ and the control $v_s$ for the linear system (6), defined by Theorem 6. Then, $y_s$ satisfies the following bound

$$\begin{aligned}&\Vert y_s\Vert _{\mathcal {C}^0([0,T]; H^1(\Omega ))} + \Vert (y_s)_t\Vert _{\mathcal {C}^0([0,T];L^2(\Omega ))} \nonumber \\&\quad \leqslant C \Big (\Vert B\Vert _{L^2(Q_T)} + e^{cs} \Vert \rho (s) B\Vert _{L^2(Q_T)} + e^{(c-1)s} \Vert (u_0)_x\Vert _{L^2(\Omega )} \nonumber \\&\qquad + se^{(c-1)s} \Vert u_0\Vert _{L^2(\Omega )} + s^{1/2}e^{(c-1)s} \Vert u_1\Vert _{L^2(\Omega )} \Big ), \end{aligned}$$

(20)

where $C>0$ is a constant that does not depend on $s\geqslant s_0$, and $c=\Vert \phi \Vert _{L^\infty (Q_T)}$.

Proof

It is well-known that for given data $(u_0, u_1) \in \varvec{V}$ and $B \in L^2(Q_T)$, we have

$$\begin{aligned}&\Vert y_s\Vert _{\mathcal {C}^0([0,T]; H^1(\Omega ))} + \Vert (y_s)_t\Vert _{\mathcal {C}^0([0,T];L^2(\Omega ))} \nonumber \\&\quad \leqslant C \left( \Vert B\Vert _{L^2(Q_T)} + \Vert (v_s)_t\Vert _{L^2(0,T)}+ \Vert (u_0)_x\Vert _{L^2(\Omega )} + \Vert u_1\Vert _{L^2(\Omega )} \right) . \end{aligned}$$

(21)

Then, using the estimate (19) from Theorem 7, we obtain (20). $\square$

3 Controllability result for the semilinear problem with ${f\in \mathcal {C}^0(\mathbb R)}$: a Schauder fixed point argument

For any $s\geqslant s_0$ and $R>0$, we introduce the class $\mathcal {C}_R(s)$, defined as the closed convex subset of $L^{\infty }(Q_T)$

$$\begin{aligned} \mathcal {C}_{R}(s): = \Big \{ \widehat{y}\in L^{\infty }(Q_T) : \ \Vert \widehat{y}\Vert _{L^{\infty }(Q_T)} \leqslant R, \ \ \Vert \rho (s) \widehat{y}\Vert _{L^2(Q_T)} \leqslant R^{1/2} \Big \} \end{aligned}$$

(22)

and assume that that the nonlinear function $f\in \mathcal {C}^0(\mathbb R)$ in (1) satisfies the growth assumption $(\mathbf{H}_\mathbf{2})$ for some $\beta ^\star$ positive precisely chosen later.

Then, for $T>2$, $s\geqslant s_0$ (to be fixed later) and for all , we first solve the linearized boundary control problem, given by

$$\begin{aligned} \left\{ \begin{aligned}&y_{tt} -y_{xx} = - f(\widehat{y})&\text {in } Q_T, \\&y(0,\cdot ) =0, \ \ y(1,\cdot ) = v&\text {in } (0,T), \\&(y(\cdot ,0),y_t(\cdot ,0))=(u_0,u_1)&\text {in } \Omega \end{aligned} \right. \end{aligned}$$

(23)

with v such that $(y(\cdot ,T), y_t(\cdot ,T))=(0,0)$ in $\Omega$. The existence of a controlled trajectory $y\in L^\infty (Q_T)$ is guaranteed by Theorem 7 with the source term $-f(\widehat{y})\in L^2(Q_T)$. Now, our aim is to prove that there exists a fixed point of the following operator

$$\begin{aligned} \Lambda _s: L^\infty (Q_T)\mapsto L^\infty (Q_T), \qquad \Lambda _s(\widehat{y})=y. \end{aligned}$$

(24)

Note that, $\Lambda _s(\widehat{y})= \Lambda ^0_s(-f(\widehat{y}), u_0, u_1)$, as per the definition (13).

Claim: We are going to show that

• 1. for $\beta ^\star >0$ small enough, there exist $R>0$ large enough and $s\geqslant s_0$ such that $\mathcal {C}_R(s)$ is stable under the map $\Lambda _s$; see Sect. 3.2;

• 2. $\Lambda _s(\mathcal {C}_R(s))$ is relatively compact subset of $\mathcal {C}_R(s)$ for the norm $\Vert \cdot \Vert _{L^\infty (Q_T)}$; see Sect. 3.3;

• 3. $\Lambda _s$ is a continuous map in $\mathcal {C}_R(s)$ for the topology induced by the norm $\Vert \cdot \Vert _{L^\infty (Q_T)}$; see Sect. 3.4.

Accordingly, by the Schauder fixed point theorem, there will exist a fixed point of $\Lambda _s$, denote by y, which will be the controlled trajectory for our semilinear problem (1).

3.1 Estimate of $\Vert \Lambda _s(\widehat{y})\Vert _{L^\infty (Q_T)}$

We begin with the following lemma.

Lemma 2

Assume $T>2$ and that there exists $\beta ^\star \geqslant 0$ such that $f\in \mathcal {C}^0(\mathbb R)$ satisfies $(\mathbf{H}_\mathbf{2})$. For any $s\geqslant s_0$ and $\widehat{y}\in L^\infty (Q_T)$, the quantity $f(\widehat{y})$ satisfies the following estimates:

$$\begin{aligned} \Vert f(\widehat{y})\Vert _{L^{2}(Q_T)}&\leqslant \alpha _1 T + \left( \alpha _2 + \beta ^\star \ln ^{3/2}_+ \Vert \widehat{y}\Vert _{L^{\infty }(Q_T)} \right) e^{cs} \Vert \rho (s)\widehat{y}\Vert _{L^2(Q_T)} , \\ \Vert \rho (s) f(\widehat{y})\Vert _{L^2(Q_T)}&\leqslant \alpha _1 T e^{-s}+ \left( \alpha _2 + \beta ^\star \ln ^{3/2}_+ \Vert \widehat{y}\Vert _{L^{\infty }(Q_T)} \right) \Vert \rho (s)\widehat{y}\Vert _{L^2(Q_T)}, \end{aligned}$$

with $c=\Vert \phi \Vert _{L^\infty (Q_T)}$.

Proof

The proof of above lemma follows from the growth assumption $(\mathbf{H}_\mathbf{2})$ on f. Observe that

$$\begin{aligned}&\left( \int _{Q_T}|f(\widehat{y})|^2 \right) ^{1/2} \leqslant \alpha _1 T+ \Vert \rho ^{-1}(s)\Vert _{\infty } \left( \int _{Q_T}|\rho (s) \widehat{y}|^2 \left( \alpha _2 + \beta ^\star \ln ^{3/2}_+ |\widehat{y}| \right) ^2 \right) ^{1/2} \\&\quad \leqslant \alpha _1 T+ \big ( \alpha _2 + \beta ^\star \ln ^{3/2}_+ \Vert \widehat{y}\Vert _{L^\infty (Q_T)} \big ) e^{cs} \Vert \rho (s) \widehat{y}\Vert _{L^2(Q_T)}, \end{aligned}$$

where we have used that $\Vert \rho ^{-1}(s)\Vert _{L^\infty (Q_T)}= \Vert e^{s\phi }\Vert _{L^\infty (Q_T)}\leqslant e^{cs}$. The other estimate is obtained in a similar way. $\square$

Proposition 1

Under the assumptions of Lemma 2, for $s\geqslant s_0$ and for all $\widehat{y}\in L^\infty (Q_T)$, the solution $y=\Lambda _s(\widehat{y})$ to the linearized system (23) satisfies the following estimates:

$$\begin{aligned}&\Vert \rho (s) y \Vert _{L^2(Q_T)} + s^{-1/2} \Vert \rho _1(s) v\Vert _{L^2(0,T)}\nonumber \\&\quad \leqslant C s^{-3/2}\Big (\alpha _2 + \beta ^\star \ln ^{3/2}_+ \Vert \widehat{y}\Vert _{L^{\infty }(Q_T) }\Big ) \Vert \rho (s)\widehat{y}\Vert _{L^2(Q_T)} \nonumber \\&\qquad +C s^{-3/2}\alpha _1 T e^{-s} +C s^{-1/2} e^{-s} \Vert u_0\Vert _{L^2(\Omega )} + C s^{-3/2} e^{-s} \Vert u_1 \Vert _{L^2(\Omega )} , \end{aligned}$$

(25)

$$\begin{aligned}&\Vert \rho (s) y_t\Vert _{L^2(Q_T)} + s^{-1/2} \Vert \rho _1(s) v_t\Vert _{L^2(0,T)}\nonumber \\&\quad \leqslant C s^{-1/2} \Big (\alpha _2+\beta ^\star \ln ^{3/2}_+ \Vert \widehat{y}\Vert _{L^\infty (Q_T)}\Big ) \Vert \rho (s) \widehat{y}\Vert _{L^2(Q_T)} \nonumber \\&\quad \quad +C s^{-1/2}\alpha _1 T e^{-s} + C s^{-1/2} e^{-s}\Vert (u_0)_x\Vert _{L^2(\Omega )} + C s^{1/2}e^{-s}\Vert u_0 \Vert _{L^2(\Omega )}\nonumber \\&\qquad + C e^{-s}\Vert u_1\Vert _{L^2(\Omega )} . \end{aligned}$$

(26)

Moreover, $y\in \mathcal {C}^0([0,T];H^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$ and

$$\begin{aligned}&\Vert y\Vert _{L^\infty (Q_T)} \leqslant C \alpha _1 T + C\alpha _1 T e^{(c-1)s} \nonumber \\&\quad + C \Big (\alpha _2 + \beta ^\star \ln ^{3/2}_+\Vert \widehat{y}\Vert _{L^\infty (Q_T)} \Big ) e^{cs} \Vert \rho (s)\widehat{y}\Vert _{L^2(Q_T)} + Ce^{(c-1)s} \Vert (u_0)_x\Vert _{L^2(\Omega )} \nonumber \\&\quad + Cse^{(c-1)s} \Vert u_0\Vert _{L^2(\Omega )} + Cs^{1/2}e^{(c-1)s} \Vert u_1\Vert _{L^2(\Omega )}. \end{aligned}$$

(27)

Proof

Put $B= -f(\widehat{y})$ in the linear model (6). Then, the proof is followed as a consequence of Theorems 6, 7, Lemma 1 and Theorem 2. $\square$

3.2 Stability of the class $\mathcal {C}_R(s)$ for suitable choices of parameters

We express the result in terms of the following lemma. We hereby recall the set $\mathcal {C}_R(s)$ defined in (22).

Lemma 3

Under the assumptions of Lemma 2, if $\beta ^\star$ in $(\mathbf{H}_\mathbf{2})$ is small enough, there exists an s and $R>0$ large enough, such that we have

$$\begin{aligned} \Lambda _s\big (\mathcal {C}_R(s) \big ) \subset \mathcal {C}_R(s) \end{aligned}$$

where $\mathcal {C}_R(s)$ is the class given in (22).

Proof

We start with any $\widehat{y}\in \mathcal {C}_R(s)$ for $s\geqslant s_0\geqslant 1$ and we look for the bounds of the solution $y=\Lambda _s(\widehat{y})$ (to (23)) with respect to the associated norms. Since $\widehat{y}\in \mathcal {C}_R(s)$, one has $\Vert \rho (s)\widehat{y}\Vert _{L^2(Q_T)}\leqslant R^{1/2}$ and $\Vert \widehat{y}\Vert _{L^\infty (Q_T)}\leqslant R$. Therefore, the estimate (25) yields

$$\begin{aligned}&\Vert \rho (s) y \Vert _{L^2(Q_T)} \leqslant C s^{-3/2}\Big (\alpha _2 + \beta ^\star \ln ^{3/2}_+ R\Big ) R^{1/2} + C s^{-3/2}\alpha _1 T e^{-s} \nonumber \\&\quad +C s^{-1/2} e^{-s} \Vert u_0\Vert _{L^2(\Omega )} + C s^{-3/2} e^{-s} \Vert u_1 \Vert _{L^2(\Omega )}. \end{aligned}$$

(28)

Similarly, estimate (27) implies

$$\begin{aligned}&\Vert y\Vert _{L^\infty (Q_T)} \leqslant C \alpha _1 T + C\alpha _1 T e^{(c-1)s} + C \Big (\alpha _2 + \beta ^\star \ln ^{3/2}_+R \Big ) e^{cs} R^{1/2} \nonumber \\&\quad + Ce^{(c-1)s} \Vert (u_0)_x\Vert _{L^2(\Omega )} + Cse^{(c-1)s} \Vert u_0\Vert _{L^2(\Omega )} + Cs^{1/2}e^{(c-1)s} \Vert u_1\Vert _{L^2(\Omega )}.\qquad \quad \end{aligned}$$

(29)

We now fix the parameter s in terms of R as follows :

$$\begin{aligned} s= \frac{1}{32 c} \ln _+R, \quad \text { with } \ c=\Vert \phi \Vert _{L^\infty (Q_T)}>1, \end{aligned}$$

(30)

where $R>0$ is chosen large enough to ensure $s\geqslant s_0\geqslant 1$. With this choice of s, the solution $y=\Lambda _s(\widehat{y})$ satisfies, in view of (28) and the fact that $\hat{y}$ belongs to $\mathcal {C}_R(s)$,

$$\begin{aligned}&\Vert \rho (s)y\Vert _{L^2(Q_T)} \leqslant \frac{C(32 c)^{3/2}}{\ln ^{3/2}_+R} \big (\alpha _2 + \beta ^\star \ln ^{3/2}_+ R\big ) R^{1/2} + \frac{C\alpha _1 T(32 c)^{3/2}}{R^{{1}/{32c}}\ln ^{3/2}_+R} \nonumber \\&\quad +\frac{C\sqrt{32c}}{ R^{{1}/{32c}}\ln ^{1/2}_+R} \Vert u_0\Vert _{L^2(\Omega )} + \frac{C(32c)^{3/2}}{ R^{1/32c}\ln ^{3/2}_+R} \Vert u_1\Vert _{L^2(\Omega )}. \end{aligned}$$

(31)

Thus, if $\beta ^\star >0$ is small enough such that

$$\begin{aligned} C (32c)^{3/2} \beta ^\star <1/4, \end{aligned}$$

(32)

it can be guaranteed for large enough $R>0$ that

(33)

involving, in view of 31 that $\Vert \rho (s) y\Vert _{L^2(Q_T)} \leqslant R^{1/2}$.

Similarly, in view of (29) and the fact that $\widehat{y}$ belongs to $\mathcal {C}_R(s)$,we infer that

$$\begin{aligned} \Vert y\Vert _{L^\infty (Q_T)}\leqslant & {} C\alpha _1 T + C\alpha _1 T R^{\left( \frac{1}{32} -\frac{1}{32c}\right) }+ C \big (\alpha _2 + \beta ^\star \ln ^{3/2}_+R \big ) R^{1/32} R^{1/2} \nonumber \\&+ \frac{C}{32c} (\ln _+R) R^{\left( \frac{1}{32} -\frac{1}{32c}\right) } \Vert u_0\Vert _{L^2(\Omega )} + C R^{\left( \frac{1}{32} - \frac{1}{32c}\right) } \Vert (u_0)_x\Vert _{L^2(\Omega )} \nonumber \\&+ \frac{C}{\sqrt{32c}} (\ln ^{1/2}_+ R) R^{\left( \frac{1}{32} - \frac{1}{32c}\right) } \Vert u_1\Vert _{L^2(\Omega )}. \end{aligned}$$

(34)

Taking $\beta ^\star >0$ as before and R large enough, we infer that (recall that $c=\Vert \phi \Vert _{L^\infty (Q_T)}>1$ so that $0<\frac{1}{32} -\frac{1}{32c}<1$)

(35)

implying from (34) that $\Vert y\Vert _{L^\infty (Q_T)} \leqslant R$. It follows that $y=\Lambda _s(\widehat{y})\in \mathcal {C}_R(s)$. This concludes the proof. $\square$

Remark 5

The smallness condition on $\beta ^\star$ is explicit:

$$\begin{aligned} \beta ^\star < \frac{1}{4C(32c)^{3/2} }, \end{aligned}$$

(36)

where C is the constant appearing in Proposition 1.

Remark 6

Provided we impose the relation (30), the above proof shows that $\mathcal {C}_R(s)$ is stable for $\Lambda _s$ for any $R \geqslant R_0$ (equivalently $s \geqslant s_0$) for a suitably large $R_0$ (equivalently $s_0$). With the above choices, in view of (33)–(35), the lower bound $R_0$ depends on $\Vert (u_0,u_1)\Vert _{\varvec{V}}$ as a power of $\Vert (u_0,u_1)\Vert _{\varvec{V}}$, so that the lower bound $s_0$ can be chosen as depending logarithmically on $\Vert (u_0,u_1)\Vert _{\varvec{V}}$. Note also that there is no upper bound for R so that the parameter s (appearing notably in the definition of the weights) can be taken arbitrarily large.

3.3 Relative compactness of the set $\Lambda _s(\mathcal {C}_R(s))$

Proposition 2

Under the assumptions of Lemma 3, $\Lambda _s(\mathcal {C}_R(s))$ is a relatively compact subset of $\mathcal {C}_R(s)$ for the $L^\infty (Q_T)$ norm.

Proof

Let $(y_n)_{n\in \mathbb N}$ be a bounded sequence in $\Lambda _s(\mathcal {C}_R(s))$. We prove that there exists a subsequence $(y_{n_k})_{k\in \mathbb N}$ of $(y_n)_{n\in \mathbb N}$ that converges strongly to some $y\in \mathcal {C}_R(s)$ with respect to the $L^\infty (Q_T)$ norm.

Thanks to Lemma 3, $(y_n)_{n\in \mathbb N}$ is a bounded sequence in $\mathcal {C}_R(s)$ and so, there exists a subsequence $y_{n_k}\in \mathcal {C}_R(s)$ and $y\in \mathcal {C}_R(s)$ such that

$$\begin{aligned} y_{n_k} \rightharpoonup ^{\star } y \ \ \text {weakly}^\star \hbox { in } L^\infty (Q_T) , \text { as } k\rightarrow +\infty . \end{aligned}$$

(37)

Now, since $(y_{n_k})_{k\in \mathbb N} \subset \Lambda _s(\mathcal {C}_R(s))$, there is a sequence $(\widehat{y}_{n_k})_{k\in \mathbb N}\subset \mathcal {C}_R(s)$ such that $y_{n_k}= \Lambda _s(\widehat{y}_{n_k})$, $\forall k\in \mathbb N$. More precisely, there exists a sequence $(v_{n_k})_{k\in \mathbb N} \in H^1_0(0,T)$ such that, for all $k\in \mathbb N$, $y_{n_k}$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&(y_{n_k})_{tt} - (y_{n_k})_{xx} = -f(\widehat{y}_{n_k})&\text {in } Q_T, \\&y_{n_k}(0,t) =0, \ \ y_{n_k}(1,t) = v_{n_k}(t)&\text {in } (0,T), \\&(y_{n_k}(\cdot ,0),(y_{n_k})_t(\cdot ,0))=(u_0,u_1)&\text {in } \Omega . \end{aligned} \right. \end{aligned}$$

Moreover, for some $C_1>0$, we have

$$\begin{aligned} \Vert y_{n_k}\Vert _{\mathcal {C}^0([0,T]; H^1_0(\Omega ))} \leqslant C_1 R, \ \ \ \Vert (y_{n_k})_t\Vert _{\mathcal {C}^0([0,T];L^2(\Omega ))} \leqslant C_1 R , \end{aligned}$$

thanks to their estimates in (20) for $B=-f(\widehat{y}_{n_k})$ and the analysis in Lemma 3. Since the embedding $\{y\in L^{\infty }(0,T; H^1_0(\Omega ) ) \mid y_t \in L^{\infty }(0,T; L^2(\Omega )) \big \} \hookrightarrow \mathcal {C}^0(\overline{Q_T})$ is compact (see [35, Corollary 8 p. 90 and Lemma 12 p. 91]), this ensures the strong convergence of $(y_{n_k})_{k\in \mathbb N}$ in $\mathcal {C}^0(\overline{Q_T})$ as $k\rightarrow +\infty$. $\square$

3.4 Continuity of the map $\Lambda _s$ in $\mathcal {C}_R(s)$

We prove the following result.

Proposition 3

Under the assumptions and result of Lemma 3, the map $\Lambda _s:\mathcal {C}_R(s) \rightarrow \mathcal {C}_R(s)$ is continuous with respect to the $L^\infty (Q_T)$ norm.

Proof

Let $(\widehat{y}_n)_{n\in \mathbb N}$ be a sequence in $\mathcal {C}_R(s)$ such that $\widehat{y}_n \rightarrow \widehat{y}$ as $n\rightarrow +\infty$ w.r.t. the $L^\infty (Q_T)$ norm for some $\widehat{y}\in \mathcal {C}_R(s)$.

Let $y_n= \Lambda _s(\widehat{y}_n)$ and prove that $y_n \rightarrow y:=\Lambda _s(\widehat{y})$ as $n\rightarrow +\infty$ w.r.t. the same norm. Since $f\in \mathcal {C}^0(\mathbb R)$, f is uniformly continuous in $[-R,R]$ implying that

$$\begin{aligned} f(\widehat{y}_n) \rightarrow f(\widehat{y}) \ \text { in } \ L^\infty (Q_T) , \ \text { as }\ n\rightarrow +\infty \end{aligned}$$

(38)

and thus $f(\widehat{y}_n) \rightarrow f(\widehat{y})$ in $L^2(Q_T)$ as $n\rightarrow +\infty$.

Now (as mentioned in Theorem 6), $\Lambda _s(\widehat{y}_n)= \Lambda ^0_s (-f(\widehat{y}_n), u_0, u_1)$ is linear continuous map from $L^2(Q_T)\times H^1_0(\Omega )\times L^2(\Omega )$ to $L^2(Q_T)$. Thus, combining the estimates (14) and (20) with $B= f(\widehat{y}_n)-f(\widehat{y})$, we get that

$$\begin{aligned} \Vert y_n -y \Vert _{L^\infty (Q_T)} \leqslant C_1 \Vert \rho (s)(f(\widehat{y}_n) -f(\widehat{y}))\Vert _{L^2(Q_T)}\leqslant C_1e^{-s} \Vert f(\widehat{y}_n) -f(\widehat{y})\Vert _{L^2(Q_T)} \end{aligned}$$

where the constant $C_1$ depends only on R and $y=\Lambda _s(\widehat{y})$. Consequently, $y_n\rightarrow y$ as $n\rightarrow +\infty$ in $L^\infty (Q_T)$. $\square$

3.5 Proof of the first item of Theorem 4

Taking $\beta ^\star$ small enough (see (32)) so that Lemma 3 applies, with s and R given by (30), we can apply Schauder fixed point theorem to $\Lambda _s$ on $\mathcal {C}_R(s)$: there exists $y_s\in \mathcal {C}_R(s)\subset L^\infty (Q_T)$ such that $y_s=\Lambda _s(y_s)$. By construction of $\Lambda _s$, there exists a function $v\in H_0^1(0,T)$ such that $y_s$ is the solution of the null controllability problem (23) with $\hat{y}=y_s$: it follows that this element $y_s$ is a controlled solution of the semilinear wave Eq. (1).

4 Construction of control by Banach fixed point approach with $f\in \mathcal {C}^1(\mathbb R)$: proof of the second item of Theorem 4

In this section, we assume that f is locally Lipschitz continuous and that $f^\prime$ satisfies $(\mathbf{H}_\mathbf{2}^\prime )$ with $\beta ^\star$ small as before. Remark that condition $(\mathbf{H}_\mathbf{2}^\prime )$ implies the condition $(\mathbf{H}_\mathbf{2})$ used in the previous section.

We endow the convex set $\mathcal {C}_R(s)$ with the distance d defined by $d(y,z)=\Vert \rho (s)(y-z)\Vert _{L^2(Q_T)}$. We easily check that $(\mathcal {C}_R(s),d)$ is a complete space. In the next proposition, we prove that the operator $\Lambda _s:\mathcal {C}_R(s)\rightarrow \mathcal {C}_R(s)$ is a contracting mapping leading to constructive method to find its fixed point.

Proposition 4

Assume that f satisfies $(\mathbf{H}_\mathbf{2}^\prime )$ with $\beta ^\star$ satisfying (36), R and s as chosen in Lemma 3. Then, for any $\widehat{y}_1, \widehat{y}_2 \in \mathcal {C}_R(s)$,

$$\begin{aligned} d(\Lambda _s(\widehat{y}_2),\Lambda _s(\widehat{y}_1)) \leqslant \frac{1}{2} d(\widehat{y}_2,\widehat{y}_1). \end{aligned}$$

(39)

In particular, $\Lambda _s$ is a contraction mapping from $\mathcal {C}_R(s)$ into itself.

Proof

Let $\widehat{y}_1, \widehat{y}_2 \in \mathcal {C}_R(s)$. From (14), we get that

$$\begin{aligned} \Vert \rho (s)(\Lambda _s (\widehat{y}_2) -\Lambda _s(\widehat{y}_1)) \Vert _{L^2(Q_T)} \leqslant C s^{-3/2} \Vert \rho (s)(f(\widehat{y}_2) - f(\widehat{y}_1))\Vert _{L^2(Q_T)} . \end{aligned}$$

Then, we can use $(\mathbf{H}_\mathbf{2}^\prime )$ to deduce

$$\begin{aligned}&\Vert \rho (s)(\Lambda _s (\widehat{y}_2) -\Lambda _s(\widehat{y}_1)) \Vert _{L^2(Q_T)}\nonumber \\&\quad \leqslant C (32 c)^{3/2} \left( \frac{\alpha }{\ln ^{3/2}_+R} + \beta ^\star \right) \Vert \rho (s)(\widehat{y}_2 - \widehat{y}_1)\Vert _{L^2(Q_T)} , \end{aligned}$$

(40)

since s is given by (30). Since $C (32c)^{3/2}\beta ^\star \leqslant 1/4$, the result follows as soon as R is large enough. $\square$

As as corollary of the previous result and the classical Banach-Picard’s fixed point theorem, the contraction property of the operator $\Lambda _s$ for $\beta ^\star$ small enough given in (32) and s and R given by (30) allows to define a convergent sequence $(y_k,v_k)_{k\in \mathbb {N}}$ to a controlled pair for (1) and prove the following precise version of the second item of Theorem 4.

Theorem 8

Let $(u_0, u_1) \in \varvec{V}$. Assume that f is locally Lipschitz continuous and satisfies $(\mathbf{H}_\mathbf{2}^\prime )$ with $\beta ^\star$ satisfying (36), s and R as chosen in Lemma 3. Then, for any $y_0 \in \mathcal {C}_R(s)$, the sequence $(y_k)_{k\in \mathbb {N}^\star } \subset \mathcal {C}_R(s)$ given by

$$\begin{aligned} y_{k+1}=\Lambda _s(y_k), \quad k\geqslant 0 , \end{aligned}$$

(where $\Lambda _s$ is defined by (24)) together with the associated sequence of controls $(v_k)_{k\in \mathbb {N}^\star }\subset H^1_0(0,T)$ strongly converges in $L^2(Q_T)\times L^2(0,T)$ to a controlled solution for (1). Moreover, the convergence is at least linear with respect to the distance d.

Proof

The convergence of the sequence $(y_k)_{k\in \mathbb {N}}$ toward $y=\Lambda _s(y)\in \mathcal {C}_R(s)$ with linear rate follows from the contraction property of $\Lambda _s$:

$$\begin{aligned} \Vert \rho (s)(y-y_{k})\Vert _{L^2(Q_T)}&= \Vert \rho (s)(\Lambda _s(y)-\Lambda _{s}(y_{k-1})) \Vert _{L^2(Q_T)} \\&\leqslant \frac{1}{2^{k}} \Vert \rho (s)(y-y_0) \Vert _{L^2(Q_T)}\leqslant \frac{1}{2^{k}} \left( R^{1/2} + e^{-s} \Vert y_0\Vert _{L^2(Q_T)} \right) . \end{aligned}$$

Let now $v\in H_0^1(0,T)$ be associated with y so that $y-y_k$ satisfies, for every $k\in \mathbb {N}^\star$

$$\begin{aligned} \left\{ \begin{aligned}&(y-y_k)_{tt} - (y-y_k)_{xx} = -\big ( f(y)- f(y_{k-1})\big )&\text {in } Q_T , \\&(y-y_k)(0,\cdot ) =0, \ \ (y-y_k)(1,\cdot ) = (v-v_k)&\text {in } (0,T), \\&((y-y_k)(\cdot ,0),(y-y_k)_t(\cdot ,0)) =(0,0)&\text {in } \Omega , \\&((y-y_k)(\cdot ,T),(y-y_k)_t(\cdot ,T)) =(0,0)&\text {in } \Omega . \end{aligned} \right. \end{aligned}$$

Estimate (14) then implies (recall $s= \frac{1}{32c} \ln _+R$)

$$\begin{aligned} \Vert \rho _1(s) (v-v_k ) \Vert _{L^2(0,T)}&\leqslant C s^{-1} \Vert \rho (s) \big ( f(y)-f(y_{k-1}) \big )\Vert _{L^2(Q_T)} \\&\leqslant C \frac{32 c}{\ln _+R} \big (\alpha + \beta ^\star \ln ^{3/2}_+R \big ) \Vert \rho (s) (y-y_{k-1})\Vert _{L^2(Q_T)} \end{aligned}$$

and therefore the convergence is at a linear rate of the sequence $(v_k)_{k\in \mathbb {N}^\star }$ toward an exact control for (1). $\square$

Remark 7

It can be observed from (40) that the constant appearing in front of $\Vert \rho (s)(\widehat{y}_2 -\widehat{y}_1)\Vert _{L^2(Q_T)}$ is getting smaller as R (consequently s) getting larger. In particular, if f satisfies

$$\begin{aligned} \lim \limits _{|r|\rightarrow +\infty } \frac{|f^{\prime }(r)|}{\ln ^{3/2}_+|r|} =0, \end{aligned}$$

(41)

then, for any given $\epsilon >0$ (however, small), the map $\Lambda _s$ is $\epsilon$-contractive for large enough $s\geqslant s_0$. In other words, the speed of convergence of the sequence $(y_k)_{k\geqslant 1}$ introduced by Theorem 8 increases with s.

5 Numerical illustrations

We present some numerical illustrations of the convergence result given by Theorem 8 and emphasize the influence of the parameter s. More precisely, for s large enough, we compute the sequence $(y_k, v_k)_{k\in \mathbb {N}}$ solution to

$$\begin{aligned} \left\{ \begin{aligned}&y_{k,tt} - y_{k,xx} = -f(y_{k-1})&\text {in } Q_T, \\&y_k(0,\cdot ) =0, \ \ y_k(1,\cdot ) = v_k&\text {in } (0,T), \\&(y_k(\cdot ,0),(y_k)_t(\cdot ,0) )=(u_0,u_1)&\text {in } \Omega ,\\&(y_k(\cdot ,T),(y_k)_t(\cdot ,T) )=(0,0)&\text {in } \Omega , \end{aligned} \right. \end{aligned}$$

(42)

obtained through the variational formulation (12) with the source term $B=-f(y_{k-1})$. We first sketch the algorithm and then discuss some numerical experiments obtained with the software FreeFem++ (see [20]).

5.1 Construction of the sequence $(y_k, v_k)_{k\geqslant 1}$

Starting with some suitable initial guess $y_0\in \mathcal {C}_R(s)$, we can obtain the solution $y_k$ to (42) with a control $v_k$ based on Theorem 6. Assume that the value of the Carleman parameter s satisfies Lemma 3. Then, for each $k\geqslant 1$, we define the unique solution $w_k\in P_s$ (see Theorem 6) of

$$\begin{aligned} (w_k,z)_{P,s}= \int _{\Omega } u_1z(\cdot ,0)\,\text {d}x-\int _\Omega u_0 z_t(\cdot ,0)\,\text {d}x-\int _{Q_T}f(y_{k-1}) z \,\text {d}x\text {d}t\quad \forall z\in P_s,\nonumber \\ \end{aligned}$$

(43)

then we set $y_k = \rho ^{-2}(s) Lw_k$ in $Q_T$ and $v_k = s\eta ^2\rho ^{-2}_1(s) (w_k)_x(1,\cdot )$ in (0, T).

The numerical approximation of the variational formulation (43) has been addressed in [8, 13] and more recently in [2]. A conformal finite dimensional approximations, say $P_{s,h}$ of $P_s$, leads to a strong convergent approximation $w_{k,h}$ of $w_k$ for the $P_s$ norm as the discretization parameter h goes to 0. Then, from $w_{k,h}$, we can define the approximated controlled solution $y_{k,h} := \rho ^{-2}(s) Lw_{k,h}$ and $v_{k,h}: = s\eta ^2\rho ^{-2}_1(s) (w_{k,h})_x(1,\cdot )$. In our semilinear setting, we shall employ an equivalent but different formulation, more appropriate for numerical purposes. First, in order to avoid the possible numerical blow up—for s large—of the terms $\rho ^{-2}(s)$ and $\rho _1^{-2}(s)$ appearing in the formulation and of order $\mathcal {O}(e^{2s})$, we introduce a change of variable. Second, in order to avoid second differentiation in order to compute $y_k$ from the definition $y_k=\rho ^{-2}(s)(w_{k,tt}-w_{k,xx})$, we incorporate directly the controlled state solution in the formulation. We refer to [31, Section 3.2] where this normalization procedure has been employed in the context of the heat equation. Precisely, we introduce the variables

$$\begin{aligned} m_k= \rho ^{-1}_1(s) w_k , \ \ \ p_k = \rho ^{-1}(s) L w_k \quad \text {in}\quad Q_T \end{aligned}$$

(44)

so that $p_k=\rho ^{-1}(s) L(\rho _1(s)m_k)$ and $y_k = \rho ^{-1}(s) p_k$ and then replace the formulation (43) by the equivalent and well-posed following mixed formulation: find $(m_k, p_k ,\lambda _k) \in \rho ^{-1}(s) P_s \times L^2(Q_T) \times L^2(Q_T)$ solution of

(45)

for all $(\overline{m},\overline{p},\overline{\lambda })\in \rho ^{-1}(s)P_s\times L^2(Q_T)\times L^2(Q_T)$. The variable $\lambda _k$ stands as a Lagrange multiplier for the constraint $p_k-\rho ^{-1}(s)L^\star (\rho _1(s) m_k)=0$ in $Q_T$. We check the following inequality

$$\begin{aligned}&\rho ^{-1}(s)L^\star (\rho _1(s) m_k)\\&\quad = \rho ^{-1}(s) \rho _1(s) L m_k + \rho ^{-1}(s) \partial _{tt}\rho _1(s) m_k + 2\rho ^{-1}(s) \partial _t \rho _1(s) (m_k)_t \\&\quad =A_1 L m_k + A_2 m_k + A_3 (m_k)_t, \end{aligned}$$

with

and we observe that the functions $A_i$ do not contain exponential with positive arguments. For instance, we get

$$\begin{aligned} \rho ^{-1}(s)\rho _1(s) = e^{-s \left( \phi (1,t) -\phi (x,t) \right) } , \end{aligned}$$

and recall that $\phi (1,t) -\phi (x,t) \geqslant 0$ in $Q_T$. Eventually, from the solution $(m_k,p_k,\lambda _k)$, the controlled pair $(y_k, v_k)$ can be retrieved using the formula

$$\begin{aligned} y_k=\rho ^{-1}(s)p_k , \quad v_k = s \eta ^2 \rho ^{-1}_1(s)(m_k)_x(1,\cdot ) . \end{aligned}$$

(46)

The sequence $(y_k, v_k)_{k\geqslant 1}$ is initialized with $(y_0, v_0)=(0,0)$ so that the iteration $(y_1,v_1)$ is the solution to the linear system (42) with the right hand side $B=-f(y_0)=-f(0)$. We perform the iterations until the following criterion (based on Proposition 4) is fulfilled

$$\begin{aligned} \frac{ \Vert \rho (s)y_{k+1} -\rho (s) y_k\Vert _{L^2(Q_T)} }{\Vert \rho (s)y_k\Vert _{L^2(Q_T)}} \leqslant 10^{-6} . \end{aligned}$$

(47)

We denote by $k^\star$ the smallest integer k such that (47) holds. Remark, since the weight $\rho (s)$ are uniformly positive and bounded in $Q_T$, the convergence of the sequence $\{\rho (s)y_k\}_{k\in \mathbb {N}}$ stated in Proposition 4) also implies the convergence of the sequence $\{y_k\}_{k\in \mathbb {N}}$.

Eventually, concerning the approximation of the formulation (45), we use a conformal space-time finite element method (as addressed in [13]). We introduce a regular triangulation $\mathcal T_h$ of $Q_T$ such that $\overline{Q}_T= \bigcup _{K \in \mathcal T_h} K$. We assume that $\{\mathcal T_h\}_{h>0}$ is a regular family, where the index h is such that $h= \displaystyle \max _{K\in \mathcal T_h} diam \, (K)$. We then approximate of the variables $p_k$ and $\lambda _k$ in the space $\displaystyle P_h: =\{ p_h \in \mathcal {C}^0({\overline{Q}}_T): p_h |_{K} \in \mathbb P_1(K), \, \forall K \in \mathcal T_h\}\subset L^2(Q_T)$, where $\mathbb P_1(K)$ denotes the space of affine functions both in x and t.

On the other hand, the variable $m_k$ is approximated with the space $\displaystyle V_h: = \{v_h \in \mathcal {C}^1({\overline{Q}}_T) : v_h |_{K} \in \mathbb P(K), \, \forall K \in \mathcal T_h \} \subset \rho ^{-1}_1(s)P_s$, where $\mathbb P(K)$ denotes the composite Hsieh–Clough–Tocher $\mathcal {C}^1$ element defined for triangles. We refer to [11, p. 356] and [3] where the implementation has been discussed. We refer to [2] for the numerical analysis of the formulation (45).

5.2 Experiments

In what follows, we take $T=2.5$ and $(u_0(x),u_1(x)) = c_{u_0}(\sin (\pi x),0)$ with $x \in \Omega =(0,1)$ parametrized by the real $c_{u_0}$. Then, we define the various weight functions appearing in the Carleman inequality (10) in Sect. 2.1 as follows: we take

$$\begin{aligned} \psi (x,t) = (x+0.02)^2 - 0.9 \left( t- T/2\right) ^2 + 2, \quad \phi (x,t)=e^{\frac{1}{2}\psi (x,t)} \qquad \text {in } Q_T, \end{aligned}$$

so that $\psi \geqslant 0.5$ in $Q_T$. The weights $\rho$ and $\rho _1$ are then defined by (8). The cut-off function $\eta$ is chosen as follows:

$$\begin{aligned} \eta (t) = e^{- \frac{1}{ \left( t+10^{-6}\right) \left( T-t+10^{-6}\right) }}, \quad \forall t \in (0,T). \end{aligned}$$

Eventually, we employ a regular space-time mesh composed of 25600 triangles and 13041 vertices corresponding to the discretization parameter $h \approx 1.25 \times 10^{-2}$.

5.2.1 Nonlinear functions with growth $r \ln ^{3/2}(2+|r|)$

In the mixed formulation (45), let us first consider the semilinear function

$$\begin{aligned} f(r)= c_f r (\alpha _2+ \beta ^\star \ln ^{3/2}(2+|r|)), \quad \forall r\in \mathbb {R} \end{aligned}$$

(48)

with $\alpha _2=\beta ^\star =1$ and some $c_f\in \mathbb R^*$, so that $f(0)=0$. We check that f satisfies $(\mathbf{H}_\mathbf{2})$ and $(\mathbf{H}_\mathbf{2}^\prime )$. In this case, the source term in (45) can be rigorously written as

$$\begin{aligned} \rho _1(s) f(\rho ^{-1}(s) p_{k-1}) =c_f \rho _1(s) \rho ^{-1}(s) p_{k-1}\Big (\alpha _2 +\beta ^\star \ln ^{3/2}(2+|y_{k-1}|) \Big ). \end{aligned}$$

I. Experiments for fixed $(c_f, c_{u_0})$ w.r.t. the parameter s. Let us make the following experiments given by Table 1, 2, for some fixed parameters $c_f$ (associated with the nonlinear function) and $c_{u_0}$ (associated with the initial data). For some large parameters $c_f=5$ and $c_{u_0}=20$, it has been checked that the value $s=1$ is not large enough to imply the Banach contraction property (ensuring the convergence of the algorithm w.r.t. the criterion (47)). Then, by choosing $s\geqslant 2$, we recover the required convergence criterion (47). We provide the results in Table 1.

Table 1 $c_{u_0}=20$ ; $c_{f}=5$; $f(r)=c_f r(1+ \ln ^{3/2}(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. s

Figure 1-left depicts the evolution of the relative error $\frac{ \Vert \rho (s)y_{k+1} -\rho (s) y_k\Vert _{L^2(Q_T)} }{\Vert \rho (s)y_k\Vert _{L^2(Q_T)}}$ w.r.t. to the iteration number k for $s\in \{1,2,3,4,5\}$. In agreement with Remark 7, we observe that the decay of the error is amplified with larger values of s. We observe that the weighted norm $\Vert \rho (s)y_{k^\star }\Vert _{L^2(Q_T)}$ and $\Vert \rho _1(s)y_{k^\star }\Vert _{L^2(0,T)}$ decrease with respect to s in agreement with the fact that the weight $\rho (s)$ and $\rho _1(s)$ decreases with s (they behave like $e^{-s}$). We observe on the contrary that the norms $\Vert y_{k^\star }\Vert _{L^2(Q_T)}$ and $\Vert v_{k^\star }\Vert _{L^2(0,T)}$ of the control-state pair are not monotonous with respect to s; this is due to the fact that the weights $s\rho ^2(s)$ and $\rho _1^2(s)$ appearing in the cost $J_s$ (see (16)) are of the same order for the values of s considered. Remark that the influence of the parameter is much stronger in the parabolic situations as it makes appear unbounded weights; we refer to [17, Section 5] for numerical experiments emphasizing this phenomenon. The stopping criterion (47) used here involves the weight $\rho (s)$ as Proposition 4 guarantees as soon as s is large enough the convergence of $\{\Vert \rho (s)y_k\Vert _{L^2(Q_T)}\}_{k\in \mathbb {N}}$. This implies notably the convergence of the sequence $\{\Vert y_k\Vert _{L^2(Q_T)}\}_{k\in \mathbb {N}}$.

If we do not incorporate the weight $\rho (s)$ in the criterion, we still observe the convergence of the corresponding sequence $\{y_k\}_{k\in \mathbb {N}}$ for the $L^2(Q_T)$-norm with again an amplification of the rate as s increases (see Fig. 1-right): a precise dependence of the rate with respect to s is, however, unknown in that case.

Fig. 1

$\frac{ \Vert \rho (s)y_{k+1} -\rho (s) y_k\Vert _{L^2(Q_T)} }{\Vert \rho (s)y_k\Vert _{L^2(Q_T)}}$ (left) and $\frac{ \Vert y_{k+1} -y_k\Vert _{L^2(Q_T)} }{\Vert y_k\Vert _{L^2(Q_T)}}$ (right) w.r.t. iterations k for $(c_f,c_{u_0})=(5,20)$

Table 2 reports experiments in the unfavorable situation for which $c_f<0$. We checked that for $c_f=-5$ ($c_{u_0}=20$ as previous), the convergence is observed from $s=3$. It is noticeable that the $L^2$ norms of the solutions and the associated controls are relatively larger compare to the case of positive $c_f$ given by Table 1. The number of iterations $k^\star$ to reach convergence is also larger : for instance, with $s=3$, we get $\Vert y_{k^\star }\Vert _{L^2(Q_T)}\approx 59.75$ and $k^{\star }=20$ for $c_f=5$, while we get $\Vert y_{k^\star }\Vert _{L^2(Q_T)}\approx 3846.94$ and $k^{\star }=32$ for $c_f=-5$.

Table 2 $c_{u_0}=20$ ; $c_{f}=-5$; $f(r)=c_f r(1+ \ln ^{3/2}(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. s

II. Experiments for fixed $(s, c_{u_0})$ w.r.t. $c_f$. Hereafter, for fixed a Carleman parameter and initial data, we consider several values of $c_f$ to study the number of iterations for which the pair the solution $y_{k^\star }$ satisfies the criterion (47); Table 3 reports some values corresponding to $s=3$ and $c_{u_0}=20$; As expected, larger negative values of $c_f$ lead to larger norms of the state-control pair; we also observe that the required number of iterations $k^\star$ increases with $\vert c_f\vert$, including in the a priori more favorable case for which $c_f>0$. For larger values of $c_f$, for instance $c_f=8$, the algorithm fails to converge, somehow in agreement with the smallness assumption on $\beta ^\star$ in our Theorem 4. For $c_f=8$, the convergence is recovered by taking a larger value of s, for instance $s=4$.

Table 3 $c_{u_0}=20$ ; $s=3$; $f(r)=c_f r(1+ \ln ^{3/2}(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. $c_f$

III. Experiments for fixed $(s, c_{f})$ w.r.t. the parameter $c_{u_0}$.

We now fix the parameters s and $c_f$ and then vary the size of the initial data $u_0$ in terms of the parameter $c_{u_0}$. We give some results in Tables 4 and 5 for $(s,c_f)=(3,-2)$ and $(s,c_f)=(3,2)$, respectively. One can observe that for large $c_{u_0}$ also, the algorithm converges. The quantity C(y, v) defined by (following the estimates in (6) or Proposition 1)

$$\begin{aligned} C(y,v) = \frac{ \Vert \rho (s) y \Vert _{L^2(Q_T)} + s^{-1/2} \Vert \rho _1(s) v\Vert _{L^2(0,T)} }{s^{-3/2} \Vert \rho (s) f(y)\Vert _{L^2(Q_T)} + s^{-1/2} e^{-s} \Vert u_0\Vert _{L^2(\Omega )} } \end{aligned}$$

(49)

is uniform with respect to the quantity $c_{u_0}$ in agreement with our theoretical results.

Table 4 $(s,c_f)=(3,-2)$; $f(r)=c_f r(1+ \ln ^{3/2}(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. $c_{u_0}$

Table 5 $(s,c_f)=(3,2)$; $f(r)=c_f r(1+ \ln ^{3/2}(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. $c_{u_0}$

IV. Evolution of the controlled solutions. In this paragraph, we present some figures of the controlled solutions and the associated controls for our semilinear system. We fix $c_f=-3$ and $c_{u_0}=10$. Figure 2 depicts the controlled solutions $y_{k^*}$ for $s=1$, 5 and 9, respectively. The corresponding optimal control is given in Fig. 3-Left for $s\in \{1,3,5,9\}$. The evolution of the $L^2(\Omega )$ norm w.r.t. $t\in (0,T)$ is depicted in Fig. 3-Right. Figures 4 and 5 are concerned with the case $c_f=3$, leading to control-state pairs with lower norms. Observe also that the optimal controls are pointwise less sensitive with respect to s when $c_f$ is positive.

Fig. 2

Controlled solution $y_{k^*}$ for $c_f=-3$, $c_{u_0}=10$ and $f(r)=c_f r(1+\ln ^{3/2}(2+|r|))$; $s\in \{1,5,9\}$

Fig. 3

$c_f=-3$, $c_{u_0}=10$ and $f(r)=c_f r (1+\ln ^{3/2}(2+|r|))$; Left: Control $v_{k^*}$ w.r.t. s; Right: Evolution of $\Vert y_{k^*}(\cdot ,t)\Vert _{L^2(\Omega )}$ w.r.t. time t

Fig. 4

Controlled solution $y_{k^*}$ for $c_f=3$, $c_{u_0}=10$ and $f(r)=c_f r(1+\ln ^{3/2}(2+|r|))$; $s\in \{1,5,9\}$

Fig. 5

$c_f=3$, $c_{u_0}=10$ and $f(r)=c_f r (1+\ln ^{3/2}(2+|r|))$; Left: Control $v_{k^*}$ w.r.t. t; Right: Evolution of $\Vert y_{k^*}(\cdot ,t)\Vert _{L^2(\Omega )}$ w.r.t. t

5.3 Nonlinear functions with growth $r |\cos (r^2)| \ln ^{3/2}(2+|r|)$

In order to enhance the importance of the assumption $(\mathbf{H}_\mathbf{2}^\prime )$ on the first derivative of f, let us consider the following nonlinear function

$$\begin{aligned} f(r)= c_f r |\cos (r^2)| \big (\alpha _2 + \beta ^\star \ln ^{3/2}(2+|r|) \big ), \quad \forall r\in \mathbb {R} \end{aligned}$$

(50)

with some $c_f\in \mathbb R^*$ and $\alpha _2=\beta ^\star =1$. It satisfies the assumption $(\mathbf{H}_\mathbf{2})$ but not $(\mathbf{H}_\mathbf{2}^\prime )$. We check that for small values of $c_f$, the algorithm converges for $s=1$. For instance, in Table 6 we give some experiments for $c_f=-1$, $c_{u_0}=20$ and $s\in \{1,2,3,4,5\}$. On the other hand, with $c_f=-2$, the method fails to converge for $s\in \{1,2,3,4,5\}$, meaning that the contraction property is lost. The convergence is recovered for s larger or equal than 6, see Table 7. Moreover, the number of iterations $k^\star$ to fulfill the convergence criterion (47) increases significantly from $c_f=-1$ to $c_f=-2$, suggesting that the amplitude of the derivative of the nonlinearity is crucial in the contracting property of the operator $\Lambda _s$.

Table 6 $c_{u_0}=20$ ; $c_{f}=-1$; $f(r)= c_f r |\cos (r^2)|(1+ \ln ^{3/2}(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. s

Table 7 $c_{u_0}=20$ ; $c_{f}=-2$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. s; when $f= c_f r |\cos (r^2)|(1+ \ln ^{3/2}(2+|r|))$

5.4 Nonlinear functions with growth $r \ln ^{p}(2+|r|)$ for $p\geqslant 2$

In this section, we first consider the following form of the nonlinear function:

$$\begin{aligned} f(r) = c_f r (\alpha _2+ \beta ^\star \ln ^2(2+|r|) ) \quad \forall r\in \mathbb {R} \end{aligned}$$

(51)

which satisfies $(\mathbf{H}_\mathbf{1})$ but not $(\mathbf{H}_\mathbf{2})$ nor $(\mathbf{H}_\mathbf{2}^\prime )$. For $(c_f,c_{u_0})=(4,10)$, we have checked that the algorithm does not converge for the Carleman parameters $s=1$ and $s=2$. For the experiments, we need at least $s>2$ to fulfill the convergence criterion (47). We present some results in Table 8.

But, as soon as we increase the $L^2$-norm of the initial data $u_0$, the result is getting worse, even if we keep the value of $c_f=4$. For instance, considering $c_{u_0}=20$ is giving the convergence for $s>4$, see Table 9. In other words, the algorithm does not really fit w.r.t. large values of norms for the initial data.

Table 8 $c_{u_0}=10$ ; $c_{f}=4$; $f(r)=c_f r (1+ \ln ^2(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. s

Table 9 $c_{u_0}=20$ ; $c_{f}=4$; $f(r)=c_f r (1+ \ln ^2(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. s

We also perform some experiments for $s=3$, $c_{u_0}=10$ to see how the algorithm behaves with respect to different values of $c_f$. In Table 10, we see that the algorithm converges for the values of $c_f\in [-3,4]$. On the other hand, for the same quantities $(s,c_{u_0})=(3,10)$, we have the divergence of our method when the nonlinear parameter $c_f \leqslant -4$ or $c_f\geqslant 5$.

Table 10 $c_{u_0}=10$ ; $s=3$; $f(r)=c_f r (1+ \ln ^2(2+|r|))$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. $c_f$

Next, we make some experiments for the nonlinearities f that behave like $r \ln ^{p} |r|$ at infinity when $p>2$ and therefore does not satisfy $(\mathbf{H}_\mathbf{1})$. Below, we consider the nonlinear function

$$\begin{aligned} f_p(r) = c_f r \big (1 + \ln ^{p}(2+|r|)\big ), \quad \text {for } p>2, \quad \forall r\in \mathbb {R}. \end{aligned}$$

(52)

We refer to Table 11 for some results associated with $c_f=-2$, $c_{u_0}=10$ and $s=3$ for $p\in \{2,2.3\}$. As expected, the value $k^\star$ increases with the value of p. Moreover, for $p\geqslant 2.4$, the algorithm does not converge anymore; more precisely, the norms of the state-control pair $(y_k,v_k)_{k\in \mathbb {N}}$ blow up with respect to k, in agreement with the result in Theorem 2 that the operator $\Lambda _s$ does not enjoy a stability result for $c_{u_0}$ and $c_f$ large enough soon as p is larger than 2.

Table 11 $c_f=-2$, $s=3$, $c_{u_0}=10$; Norms of $(y_{k^\star },v_{k^\star })$ w.r.t. $f_p$ given by (52)

6 Concluding remarks

By introducing a functional in the Carleman setting different than in the seminal paper of Zuazua [41], we have derived, under similar assumptions, a somehow simpler proof of the boundary controllability of a semilinear wave equation of the form $y_{tt}-y_{xx}+f(y)=0$. Moreover, assuming an additional growth assumption on $f^\prime$, we have constructed a sequence of state-control pairs, solution of a linear boundary controllability problem, converging pointwise and with a linear rate to a solution of the semilinear equation. As in the recent work [17] devoted to the distributed controllability for a semilinear heat equation, the analysis emphasizes the role of the Carleman weights parameterized by the real s. Numerical experiments illustrate that the speed of convergence of the sequence is amplified as the Carleman parameter s is larger.

Our analysis is based on a simple fixed point strategy which consists to see the nonlinear term as a source term. It would be interesting to analyze whether or not the fixed point operator introduced by Zuazua in 1993, involving a potential, is, after reformulation in a functional Carleman setting, contracting for s large enough.

The fixed point argument employed here requires uniform bounds of the controlled trajectories for a linear wave equation: this is achieved by assuming the initial data in $H^1_0(\Omega )\times L^2(\Omega )$ and by imposing that the control satisfies at the initial and final time some compatibility conditions with the solution: this leads to boundary controls in $H_0^1(0,T)$ and then, in our one-dimensional situation, to trajectories in $L^\infty (Q_T)$. Assuming more regularity on the initial conditions, we may extend our results for multi-dimensional situations and for nonlinearities depending on the gradient of the solution. This will be addressed in future works.

Appendix: Proof of Theorem 7

In what follows, in order to simplify the notations, we shall just write $\rho =\rho (t)$ and $\rho _1=\rho _1(t)$ instead of $\rho =\rho (s;x,t)$ and $\rho _1=\rho _1(s;t)$.

Preliminary to the proof and following [18], for all $f\in \mathcal {C}^0(\mathbb R;E)$ (where E is a Banach space) and any $\tau >0$, we define $\delta _\tau f:= f\left( t+\frac{\tau }{2} \right) - f\left( t-\frac{\tau }{2} \right)$ and

$$\begin{aligned} \mathcal {T}_\tau f := \frac{1}{\tau }\delta _\tau \left( \frac{\delta _\tau f}{\tau }\right) = \frac{f(t+\tau )-2f(t)+f(t-\tau )}{\tau ^2}. \end{aligned}$$

Let now $w_s\in P_s$ and the solution $y_s \in L^2(Q_T)$ be given by 6. Then, z defined by $z=\mathcal {T}_\tau w_s$ belongs to $P_s$, where $w_s$ as well as $y_s$ can be extended uniquely on $(-\infty ,0)$ and $(T,+\infty )$. Indeed, in the interval $(-\infty , 0)$ the solution $y_s$ satisfies the following set of equations

(53)

where the source term $B\in L^2(Q_T)$ is assumed to be extendable by 0 outside (0, T). Recall that the boundary condition $y_s(1,t) =0$ holds outside (0, T) since $\eta =0$ (appearing in the formula of $v_s$) vanishes outside $(\delta , T-\delta )$.

Similarly, in $(T, + \infty )$ we can define the solution $y_s$ uniquely, and $y_s(t)=0$ for all $t\geqslant T$. It follows that the solution $y_s$ satisfies $y_s\in \mathcal {C}^0(\mathbb R;L^2(\Omega ))\cap \mathcal {C}^1(\mathbb R;H^{-1}(\Omega ))$ and $y_s\in \mathcal {C}^0((-\infty ,\delta ];H^1(\Omega ))\cap \mathcal {C}^1((-\infty ,\delta ];L^2(\Omega ))$ and $y_s\in \mathcal {C}^0([T-\delta , +\infty );H^1(\Omega ))\cap \mathcal {C}^1([T-\delta ,+ \infty );L^2(\Omega ))$ (see [26]). We extend as well the weights $\rho$ and $\rho _1$ in $\Omega \times \mathbb {R}$ so that it preserves smoothness and positivity properties.

This ensures the extension of the solution $w_s$ which satisfies the following set of equations in $\mathbb R$

(54)

Moreover, it can be seen that $Lw_s=0$ in $[T, + \infty )$, since $y_s$ is a controlled solution to (53).

We now proceed to the proof of Theorem 7, done in three steps.

Step 1 : We suppose first that $u_0\in H^1_0(\Omega )\cap H^2(\Omega )$, $u_1\in H_0^1(\Omega )$ and $B\in \mathcal {D}(0,T;L^2(\Omega ))$ and prove that $v_s\in H^1(0,T)$ and $(y_s)_t\in L^2(Q_T)$.

We start by considering the variational formulation (12) by choosing $z=\mathcal {T}_\tau w_s$ as test function. Since $w_s \in \mathcal {C}^0(\mathbb R;H^1_0(\Omega ))\cap \mathcal {C}^1(\mathbb R;L^2(\Omega ))$ solves (54), it is clear that $\mathcal {T}_\tau w_s \in \mathcal {C}^0([0,T]; H^1_0(\Omega )) \cap \mathcal {C}^1([0,T]; L^2(\Omega ))$, $(\mathcal {T}_\tau w_s)_x\in L^2(0,T)$. With this z, the formulation reads

$$\begin{aligned} \int _{Q_T}\rho ^{-2} L w_s L \mathcal {T}_\tau w_s \,\text {d}x\text {d}t+ s \int _0^T \eta ^2(t) \rho ^{-2}_1 (w_s)_x(1,t) \mathcal {T}_\tau (w_s)_x(1,t) \text {d}t\nonumber \\ = \int _\Omega u_1(x) \mathcal {T}_\tau w_s(x,0) \,\text {d}x- \int _\Omega u_0(x) \mathcal {T}_\tau (w_s)_t(x,0) \,\text {d}x+ \int _{Q_T}B \mathcal {T}_\tau w_s \,\text {d}x\text {d}t. \end{aligned}$$

(55)

Sub-step 1. Let us start with the first integral in the left hand side of (55). We have

$$\begin{aligned} \begin{aligned}&\int _{Q_T}\rho ^{-2}(t)Lw_s(t) L \mathcal {T}_\tau w_s(t) \,\text {d}x\text {d}t\\&=\frac{1}{\tau }\int _{Q_T} \rho ^{-2}(t) Lw_s(t)\frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\,\text {d}x\text {d}t\\&-\frac{1}{\tau }\int _{Q_T} \rho ^{-2}(t) Lw_s(t)\frac{L w_s(t)-Lw_s(t-\tau )}{\tau }\,\text {d}x\text {d}t\\&=\frac{1}{\tau }\int _{Q_T} \rho ^{-2}(t) Lw_s(t)\frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\,\text {d}x\text {d}t\\&-\frac{1}{\tau }\int _{-\tau }^{T-\tau }\!\!\!\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\,\text {d}x\text {d}t\\&= \int _{Q_T} \left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\\&-\frac{1}{\tau }\int _{-\tau }^{0}\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\\&{-\frac{1}{\tau }\int _T^{T-\tau }\!\!\!\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t.} \end{aligned} \end{aligned}$$

(56)

Now, observe that

$$\begin{aligned} \begin{aligned}&\int _{Q_T} \left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\\&= \int _{Q_T} \rho ^2(t)\left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \rho ^{-2}(t)\\&\times \left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\\&= - \int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\\& + \int _{Q_T} \rho ^2(t) \left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \\&\times \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau ) \,\text {d}x\text {d}t. \end{aligned} \end{aligned}$$

(57)

The equality (56) then reads

$$\begin{aligned}&\int _{Q_T}\rho ^{-2} Lw_s(t) L \mathcal {T}_\tau w_s(t) \,\text {d}x\text {d}t\nonumber \\ =&- \int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\nonumber \\&+ \int _{Q_T} \rho ^2(t) \left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \nonumber \\&\times \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau ) \,\text {d}x\text {d}t\nonumber \\&~~~~~~~~~~~~~~ -\frac{1}{\tau }\int _{-\tau }^{0}\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\nonumber \\&{-\frac{1}{\tau }\int _T^{T-\tau }\!\!\!\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t.} \end{aligned}$$

(58)

Next, we shall look into the second term in the left hand side of (55). First, recall the smooth function $\eta$ given by (9) satisfies $\eta =0$ in $(-\infty , \delta ]\cup [T-\delta , +\infty )$ (with $\delta >0$ given in (9)). Then, in a similar way that have lead to (56), we have assuming $|\tau |\leqslant \delta$ :

$$\begin{aligned}&\int _0^T \eta ^2(t)\rho _1^{-2}(t)(w_s)_x(1,t) \mathcal {T}_\tau (w_s)_x(1,t) \text {d}t\nonumber \\&\quad = \int _0^T \left( \frac{\eta ^2(t)\rho _1^{-2}(t)(w_s)_x(1,t)-\eta ^2(t+\tau )\rho _1^{-2}(t+\tau ) (w_s)_x(1,t+\tau )}{\tau }\right) \nonumber \\&\qquad \times \left( \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right) \text {d}t. \end{aligned}$$

(59)

Then, using the identity

$$\begin{aligned} ad- bc = \frac{(a-c)(b+d)-(a+c)(b-d)}{2}, \qquad \forall (a,b,c,d) \in \mathbb R^4 \end{aligned}$$

(60)

with $a = \eta ^2(t) \rho ^{-2}_1(t)$, $b = (w_s)_x(1,t+\tau )$, $c=\eta ^2(t+\tau ) \rho ^{-2}_1(t+\tau )$ and $d = (w_s)_x(1,t)$, we obtain from (59)

$$\begin{aligned}&\int _0^T \eta ^2(t) \rho _1^{-2}(t)(w_s)_x(1,t) \mathcal {T}_\tau (w_s)_x(1,t) \text {d}t\nonumber \\ =&\int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t) - \eta ^2(t+\tau )\rho _1^{-2}(t+\tau )\right) \left( (w_s)_x(1,t)+(w_s)_x(1,t+\tau ) \right) }{2\tau } \nonumber \\&\times \left( \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right) \text {d}t\nonumber \\&\qquad \quad -\int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau ) \right) }{2}\left| \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right| ^2 \text {d}t. \end{aligned}$$

(61)

Now, using (58) and (61) in the formulation (55), we have

$$\begin{aligned}&\int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\nonumber \\&+s\int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau ) \right) }{2}\left| \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right| ^2 \text {d}t\nonumber \\ =&\int _{Q_T} \rho ^2(t) \left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \nonumber \\&\qquad \times \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau ) \,\text {d}x\text {d}t\nonumber \\&~~~~~~~~~~~~~~ -\frac{1}{\tau }\int _{-\tau }^{0}\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\nonumber \\&{-\frac{1}{\tau }\int _T^{T-\tau }\!\!\!\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t}\nonumber \\&+s \int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t) - \eta ^2(t+\tau )\rho _1^{-2}(t+\tau )\right) \left( (w_s)_x(1,t)+(w_s)_x(1,t+\tau ) \right) }{2\tau } \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times \left( \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right) \text {d}t\nonumber \\&- \int _{Q_T}B \mathcal {T}_\tau w_s \,\text {d}x\text {d}t-\int _\Omega u_1 \mathcal {T}_\tau w_s(\cdot ,0) \,\text {d}x+ \int _\Omega u_0 \mathcal {T}_\tau (w_s)_t(\cdot ,0) \,\text {d}x\nonumber \\ :=&I_1 +I_2 + I_3 +I_4 +I_5 +I_6+I_7. \end{aligned}$$

(62)

Sub-step 2. In this step, we obtain precise estimates for the terms $I_1$ and ${I_4}$ and then an estimate of the left hand side of (62).

(i) Estimate of ${I_1}$. Young’s inequality leads to

$$\begin{aligned} |I_1| \leqslant&\frac{1}{2} \int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\nonumber \\&\quad + \frac{1}{2} \int _{Q_T} \rho ^2(t)\left| \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau )\right| ^2 \,\text {d}x\text {d}t. \end{aligned}$$

(63)

(ii) Estimate of ${I_4}$. We have

$$\begin{aligned} |{I_4}|&=s\left| \int _0^T \frac{\left( \eta (t)\rho _1^{-1}(t)-\eta (t+\tau )\rho _1^{-1}(t+\tau )\right) \left( (w_s)_x(1,t)+ (w_s)_x(1,t+\tau )\right) }{2\tau } \right. \nonumber \\&\left. \times \frac{\left( \eta (t)\rho _1^{-1}(t)+\eta (t+\tau )\rho _1^{-1}(t+\tau ) \right) \left( (w_s)_x(1,t+\tau )- (w_s)_x(1,t)\right) }{\tau } \text {d}t\right| \nonumber \\&\leqslant 2s\int _0^T \left| \frac{\left( \eta (t)\rho _1^{-1}(t)-\eta (t+\tau )\rho _1^{-1}(t+\tau )\right) \left( (w_s)_x(1,t)+ (w_s)_x(1,t+\tau )\right) }{2\tau } \right| ^2 \text {d}t\nonumber \\&\ \ +\frac{s}{8}\int _0^T \left| \eta (t)\rho _1^{-1}(t)+\eta (t+\tau )\rho _1^{-1}(t+\tau ) \right| ^2\left| \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right| ^2 \text {d}t\nonumber \\&\leqslant 2s\int _0^T \left| \frac{\left( \eta (t)\rho _1^{-1}(t)-\eta (t+\tau )\rho _1^{-1}(t+\tau )\right) \left( (w_s)_x(1,t)+ (w_s)_x(1,t+\tau )\right) }{2\tau } \right| ^2 \text {d}t\nonumber \\&\ \ +\frac{s}{2}\int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau ) \right) }{2}\left| \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right| ^2 \text {d}t. \end{aligned}$$

(64)

(iii) A first estimate of the left hand side of (62). The previous estimates and (62) give

$$\begin{aligned}&\frac{1}{2}\int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\nonumber \\&+ \frac{s}{2}\int _0^T\frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau )\right) }{2}\left| \frac{(w_s)_x(1,t)- (w_s)_x(1,t+\tau )}{\tau }\right| ^2 \text {d}t\nonumber \\&\leqslant \frac{1}{2} \int _{Q_T} \rho ^2(t)\left| \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau )\right| ^2 \,\text {d}x\text {d}t\nonumber \\&+ \left| {\frac{1}{\tau }}\int _{-\tau }^{0}\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\right| \nonumber \\&+{\left| \frac{1}{\tau }\int _T^{T-\tau }\!\!\!\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\right| }\nonumber \\&+2s\int _0^T\left| \frac{\left( \eta (t)\rho _1^{-1}(t)-\eta (t+\tau )\rho _1^{-1}(t+\tau )\right) \left( (w_s)_x(1,t)+(w_s)_x(1,t+\tau ) \right) }{2\tau }\right| ^2 \text {d}t\nonumber \\&+\left| \int _{Q_T}B \mathcal {T}_\tau w_s \,\text {d}x\text {d}t\right| +\left| \int _\Omega u_0 \mathcal {T}_\tau (w_s)_t(\cdot ,0) \,\text {d}x\right| +\left| \int _\Omega u_1 \mathcal {T}_\tau w_s(\cdot ,0) \,\text {d}x\right| \nonumber \\&:=J_1+J_2+J_3+J_4+J_5+J_6+ J_7. \end{aligned}$$

(65)

Sub-step 3 : We prove that the left hand side of (65) is bounded uniformly with respect to ${|\tau |}\in [0,\delta ]$.

(i) $J_1$ is bounded. Since $\rho ^{-2}\in {\mathcal {C}^\infty (\mathbb {R}\times \overline{\Omega })}$, $(\rho ^{-1})_t=-2s\lambda \beta (t-\frac{T}{2}) \phi \rho ^{-1}$ and $Lw_s\in \mathcal {C}^0(\mathbb R;L^2(\Omega ))$:

$$\begin{aligned}&\int _{Q_T} \rho ^2(t)\left| \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau )\right| ^2 \,\text {d}x\text {d}t\\&\quad \rightarrow 4s^2\lambda ^2\beta ^2\int _{Q_T} \biggl (t-\frac{T}{2}\biggr )^2 \phi ^2(t)\rho ^{2}(t)y_s^2 \,\text {d}x\text {d}t\end{aligned}$$

as $\tau \rightarrow 0$ and thus $J_1$ is bounded.

(ii) $J_2$ is bounded. Since $\rho ^{-2}Lw_s=y_s\in \mathcal {C}^0(\mathbb R;L^2(\Omega ))$, $\rho ^{-2}\in {\mathcal {C}^\infty (\mathbb {R}\times \overline{\Omega })}$ and $L w_s=\rho ^2y_s\in \mathcal {C}^1((-\infty ,\delta ];L^2(\Omega ))$ we have, as $\tau \rightarrow 0$

$$\begin{aligned}&\left| {\frac{1}{\tau }}\int _{-\tau }^{0}\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\right| \\&\quad \rightarrow \left| \int _\Omega y_s(0)(\rho ^2y_s)_t(0)\right| \end{aligned}$$

and thus $J_2$ is bounded.

(iii) $J_3$ is bounded. Since $\rho ^{-2}Lw_s=y_s\in \mathcal {C}^0(\mathbb R;L^2(\Omega ))$, $\rho ^{-2}\in \mathcal {C}^\infty (\mathbb {R}\times \overline{\Omega })$ and $L w_s=\rho ^2y_s\in \mathcal {C}^1([T-\delta , +\infty );L^2(\Omega ))$ we have, as $\tau \rightarrow 0$

$$\begin{aligned}&\left| \frac{1}{\tau }\int _T^{T-\tau }\!\!\!\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\right| \\&\quad \rightarrow \left| \int _\Omega y_s(T)(\rho ^2y_s)_t(T)\right| =0 \end{aligned}$$

and thus $J_3$ is bounded.

(iv) $J_4$ is bounded. Since $(w_s)_x(1,\cdot )\in L^2(0,T)$ and $\eta \rho ^{-1}_1\in \mathcal {C}^1(\mathbb R)$ we have

$$\begin{aligned} \begin{aligned}&2s\int _0^T\left| \frac{\left( \eta (t)\rho _1^{-1}(t)-\eta (t+\tau )\rho _1^{-1}(t+\tau )\right) \left( (w_s)_x(1,t)+(w_s)_x(1,t+\tau ) \right) }{2\tau }\right| ^2 \text {d}t\\&\rightarrow 2s\int _0^T\left| (\eta \rho _1^{-1})_t(t) (w_s)_x(1,t)\right| ^2 \text {d}t\end{aligned} \end{aligned}$$

as $\tau \rightarrow 0$ and thus ${J_4}$ is bounded.

(v) $J_5$ is bounded. For $\tau$ small enough, since $B\in \mathcal {D}(\mathbb R;L^2(\Omega ))$ and $w_s\in \mathcal {C}{{}^0}(\mathbb R; L^2(\Omega ))$, we have

$$\begin{aligned} \left| \int _{Q_T}B \mathcal {T}_\tau w_s \,\text {d}x\text {d}t\right| = \left| \int _{-\tau }^{T+\tau }\!\!\!\int _\Omega \mathcal {T}_\tau B w_s\,\text {d}x\text {d}t\right| \rightarrow \left| \int _{Q_T}B_{tt} w_s\,\text {d}x\text {d}t\right| \end{aligned}$$

(66)

as $\tau \rightarrow 0$ and thus ${ J_5}$ is bounded.

(vi) $J_6$ is bounded. We have $L w_s=\rho ^2 y_s\in \mathcal {C}^0(\mathbb R;L^2(\Omega ))$ and $w_s\in \mathcal {C}^0(\mathbb R;H^1_0(\Omega ))$, thus $(w_s)_{tt}= Lw_s + (w_s)_{xx} \in \mathcal {C}(\mathbb R;H^{-1}(\Omega ))$. We then have, for all $t \in \mathbb R$:

$$\begin{aligned} (w_s)_t(t)- (w_s)_t(0)=\int _0^t Lw_s(\xi ) \,\text {d}\xi + \int _0^t (w_s)_{xx}(\xi )\,\text {d}\xi . \end{aligned}$$

This yields

$$\begin{aligned} \mathcal {T}_\tau (w_s)_t(0)&=\frac{(w_s)_t(\tau )-2 (w_s)_t(0)+ (w_s)_t(-\tau )}{\tau ^2}\\&=\frac{1}{\tau ^2}\bigg (\int _0^{\tau } L w_s(\xi )\,\text {d}\xi +\int _0^{-\tau } L w_s(\xi )\,\text {d}\xi \\&\quad +\int _0^{\tau } (w_s)_{xx}(\xi )\,\text {d}\xi +\int _0^{-\tau } (w_s)_{xx} (\xi )\,\text {d}\xi \bigg ) \\&=\frac{2}{\tau } \int _0^{\tau }\frac{\xi }{\tau } \bigg (\frac{L w_s(\xi )-L w_s(-\xi )}{2\xi }\bigg )\,\text {d}\xi \\&\quad + \frac{2}{\tau } \int _0^{\tau }\frac{\xi }{\tau } \bigg (\frac{ (w_s)_{xx}(\xi )- (w_s)_{xx} (-\xi )}{2\xi }\bigg ) \,\text {d}\xi . \end{aligned}$$

Now, since $L w_s=\rho ^2y_s\in \mathcal {C}^1((-\infty ,\delta ];L^2(\Omega ))$, we write that

$$\begin{aligned}&\int _\Omega u_0\frac{2}{\tau } \int _0^{\tau } \frac{\xi }{\tau } \bigg (\frac{L w_s(\xi )-L w_s(-\xi )}{2\xi } \bigg )\,\text {d}\xi \,\text {d}x\nonumber \\&\rightarrow \int _\Omega u_0(\rho ^2y_s)_t(0)\,\text {d}x=-2s\lambda \beta T\int _\Omega \phi (0)\rho ^2(0)u_0^2\,\text {d}x+ \int _\Omega \rho ^2(0) u_0u_1 \,\text {d}x\ \ \text {as } \tau \rightarrow 0. \end{aligned}$$

(67)

On the other hand, since $u_0\in H^2(\Omega )\cap H^1_0(\Omega )$ and $w_s\in \mathcal {C}^0(\mathbb R; H^1_0(\Omega ))$:

$$\begin{aligned}&\frac{2}{\tau } \int _0^\tau \left\langle \frac{\xi }{\tau } \bigg (\frac{ (w_s)_{xx}(\xi )- (w_s)_{xx} (-\xi )}{2\xi }\bigg ) , u_0 \right\rangle _{H^{-1}, H^1_0} \,\text {d}\xi \nonumber \\&= \frac{2}{\tau } \int _0^\tau \frac{\xi }{\tau } \int _\Omega (u_0)_{xx} \frac{(w_s)(\xi )-(w_s)(-\xi )}{2\xi } \,\text {d}x \,\text {d}\xi \nonumber \\&\rightarrow \int _\Omega (u_0)_{xx} (w_s)_t(0) \,\text {d}x\quad \text {as}\, \tau \rightarrow 0 \end{aligned}$$

(68)

since moreover $w_s\in \mathcal {C}^1(\mathbb {R};L^2(\Omega ))$. Thus

$$\begin{aligned} \int _\Omega u_0 \mathcal {T}_\tau (w_s)_t(\cdot ,0) \,\text {d}x&\rightarrow -2s\lambda \beta T\int _\Omega \phi (0)\rho ^2(0)u_0^2 + \int _\Omega \rho ^2(0) u_0u_1\nonumber \\&\quad +\int _\Omega (u_0)_{xx} (w_s)_t(0) \,\text {d}x \end{aligned}$$

(69)

as $\tau \rightarrow 0$ and thus ${J_6}$ is bounded.

(vii) $J_7$ is bounded. We have $L w_s=\rho ^2 y_s\in \mathcal {C}^0(\mathbb R;L^2(\Omega ))$ and $w_s\in \mathcal {C}^1(\mathbb R;H^1_0(\Omega ))$, thus $(w_s)_{tt}= \rho ^2 y_s + (w_s)_{xx} \in \mathcal {C}^0(\mathbb R;H^{-1}(\Omega ))$. Therefore

$$ \mathcal {T}_\tau w(0)=\frac{w(\tau )-2w(0)+w(-\tau )}{\tau ^2}\rightarrow (w_s)_{tt}(0)=(w_s)_{xx}(0)+\rho ^2(0)u_0\hbox { in } H^{-1}(\Omega )$$

as $\tau \rightarrow 0$ and thus

$$\begin{aligned} \int _\Omega u_1 \mathcal {T}_\tau w (\cdot ,0) \,\text {d}x\rightarrow&\langle (w_s)_{tt}(0),u_1\rangle _{H^{-1}(\Omega )\times H^1_0(\Omega )}\nonumber \\&=\langle (w_s)_{xx}(0),u_1\rangle _{H^{-1}(\Omega )\times H^1_0(\Omega )}+\int _\Omega \rho ^2(0) u_1 u_0\,\text {d}x\\&= -\int _\Omega (w_s)_{x}(\cdot ,0)(u_1)_x\,\text {d}x+\int _\Omega \rho ^2(0) u_1 u_0 \nonumber \,\text {d}x. \end{aligned}$$

(70)

as $\tau \rightarrow 0$.

${J_7}= \left| \int _\Omega u_1 \mathcal {T}_\tau w (\cdot ,0) \,\text {d}x\right|$ is therefore bounded.

(viii) Then we can conclude, from (65), that the terms

$$\begin{aligned} \int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\end{aligned}$$

and

$$\begin{aligned} \int _0^T\frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau )\right) }{2}\left| \frac{(w_s)_x(1,t)- (w_s)_x(1,t+\tau )}{\tau }\right| ^2 \text {d}t\end{aligned}$$

are bounded. Remark that this implies that the two terms $\int _{Q_T}\rho ^{-2}(t) |L (\frac{\delta _\tau w_s}{\tau })|^2 \,\text {d}x\text {d}t$ and $\int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(\frac{\delta _\tau w_s}{\tau })_x(1,t)|^2 \text {d}t$ are bounded; indeed,

$$\begin{aligned}&\int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(\frac{\delta _\tau w_s}{\tau })_x(1,t)|^2 \text {d}t\\&\leqslant 2\int _0^T\frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau )\right) }{2}\left| \frac{(w_s)_x(1,t)- (w_s)_x(1,t+\tau )}{\tau }\right| ^2 \text {d}t. \end{aligned}$$

We also have

$$\begin{aligned} \begin{aligned}&\int _{Q_T}\rho ^{-2}(t) |L (\frac{\delta _\tau w_s}{\tau })|^2 \,\text {d}x\text {d}t\\&\quad \leqslant 2\int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\\&\qquad +2\int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t)-\rho ^{-2}(t+\tau )}{\tau }Lw_s(t+\tau )\right| ^2 \,\text {d}x\text {d}t\\ \end{aligned} \end{aligned}$$

and each term of the right hand side is bounded.

Sub-step 4. In this step, we prove that $v_s\in H^1(0,T)$ and $y_s\in \mathcal {C}^0([0,T];H^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$.

Since $\frac{\delta _\tau w_s}{\tau }\in \mathcal {C}^0([0,T];H_0^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$ and satisfies $(\frac{\delta _\tau w_s}{\tau })_x(1,\cdot )\in L^2(0,T)$ then the Carleman estimates (10) gives

$$\begin{aligned}&s \int _{Q_T}\rho ^{-2}(t) \biggl (| (\frac{\delta _\tau w_s}{\tau })_{t}|^2 + | (\frac{\delta _\tau w_s}{\tau })_x|^2\biggr ) \,\text {d}x\text {d}t\\&\quad + s^3 \int _{Q_T}\rho ^{-2}(t) | \frac{\delta _\tau w_s}{\tau }|^2 \,\text {d}x\text {d}t\\&\quad + s \int _\Omega \rho ^{-2}(0) \biggl (|(\frac{\delta _\tau w_s}{\tau })_t(x,0)|^2 + |(\frac{\delta _\tau w_s}{\tau })_x(x,0)|^2\biggr ) \,\text {d}x\\&\quad + s^3 \int _\Omega \rho ^{-2}(0) |\frac{\delta _\tau w_s}{\tau }(x,0)|^2 \,\text {d}x\\&\quad \leqslant C \int _{Q_T}\rho ^{-2}(t) |L (\frac{\delta _\tau w_s}{\tau })|^2 \,\text {d}x\text {d}t+ C s \int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(\frac{\delta _\tau w_s}{\tau })_x(1,t)|^2 \text {d}t. \end{aligned}$$

Therefore, since the right hand side is bounded, $(\frac{\delta _\tau w_s}{\tau })_{t}$ and $(\frac{\delta _\tau w_s}{\tau })_x$ are bounded in $L^2(Q_T)$ and thus $(w_s)_{tt}\in L^2(Q_T)$ and $(w_s)_{t}\in L^2(0,T;H^1_0(\Omega ))$. Moreover, $\frac{\delta _\tau w_s}{\tau }(\cdot ,0)$ is bounded in $H^1_0(\Omega )$ thus $(w_s)_t(\cdot ,0)\in H^1_0(\Omega )$. We also have $L (\frac{\delta _\tau w_s}{\tau })$ bounded in $L^2(Q_T)$ so $L( w_s)_t\in L^2(Q_T)$. Thus $(w_s)_t$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&L (w_s)_t \in L^2(Q_T), \\&(w_s)_t(0,t)=(w_s)_t(1,t) =0, \quad t\in (0,T) \\&((w_s)_t (0),(w_s)_{tt}(0))\in H_0^1(\Omega )\times L^2(\Omega ) \end{aligned} \right. \end{aligned}$$

(71)

and thus $(w_s)_t\in \mathcal {C}^0([0,T];H^1_0(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$ and $(w_s)_{tx}(1,\cdot )\in L^2(0,T)$. Therefore from the definition of $v_s$, $v_s\in H^1(0,T)$ while from the equation satisfied by $(y_s,v_s)$ (see (17)), $y_s\in \mathcal {C}^0([0,T];H^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$.

Remark 1 We then have $w_s\in \mathcal {C}^1([0,T];H^1_0(\Omega ))\cap \mathcal {C}^2([0,T];L^2(\Omega ))$ and from the equation satisfied by $w_s$, since $Lw_s\in \mathcal {C}^1([0,T];L^2(\Omega ))$ we deduce that $(w_s)_{xx}=(w_s)_{tt}-L w_s \in \mathcal {C}^0([0,T];L^2(\Omega ))$ and thus that $(w_s)_{xx}(\cdot ,0)\in L^2(\Omega )$.

Step 2 : In this step, we give estimates on $(v_s)_t$ and $(y_s)_t$.

First of all, since $(w_s)_t\in \mathcal {C}^0([0;T];H^1_0(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$, $L (w_s)_t\in L^2(Q_T)$ and $(w_s)_{tx}(1,\cdot )\in L^2(0,T)$, we can write the Carleman estimate (10) for $(w_s)_t$ leading to

$$\begin{aligned}&s \int _{Q_T}\rho ^{-2}(t) (| (w_s)_{tt}|^2 + | (w_s)_{tx}|^2) \,\text {d}x\text {d}t+ s^3 \int _{Q_T}\rho ^{-2}(t) | (w_s)_t|^2 \,\text {d}x\text {d}t\nonumber \\&\quad + s \int _\Omega \rho ^{-2}(0) (|(w_s)_{tt}(x,0)|^2 + (w_s)_{tx}(x,0)|^2) \,\text {d}x+ s^3 \int _\Omega \rho ^{-2}(0) |(w_s)_t(x,0)|^2 \,\text {d}x\nonumber \\&\quad \leqslant C \int _{Q_T}\rho ^{-2}(t) |(L w_s)_t|^2 \,\text {d}x\text {d}t+ C s \int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(w_s)_{tx}(1,t)|^2 \text {d}t. \end{aligned}$$

(72)

Sub-step 1 : In this step, we pass to the limit when $\tau \rightarrow 0$ in equation (62). We have, since $y_s=\rho ^{-2} Lw_s\in \mathcal {C}^1(\mathbb {R};L^2(\Omega ))$ :

$$\begin{aligned} \int _{Q_T} \rho ^2(t)\left| \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right| ^2 \,\text {d}x\text {d}t\rightarrow \int _{Q_T} \rho ^2(t)\left| (y_s)_t\right| ^2 \,\text {d}x\text {d}t\end{aligned}$$

as $\tau \rightarrow 0$ and since $(w_s)_{tx}(1,\cdot )\in L^2(-\delta ,T+\delta )$ and $\eta \rho _1^{-1}\in \mathcal {C}(\mathbb {R})$ :

$$\begin{aligned} \begin{aligned} \int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t)+\eta ^2(t+\tau )\rho _1^{-2}(t+\tau ) \right) }{2}&\left| \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right| ^2 \text {d}t\\&\rightarrow \int _0^T \eta ^2(t)\rho _1^{-2}(t) \left| (w_s)_{tx}(1,t)\right| ^2 \text {d}t\end{aligned} \end{aligned}$$

as $\tau \rightarrow 0$. Since $y_s=\rho ^{-2} Lw_s\in \mathcal {C}^1(\mathbb R;L^2(\Omega ))$, $Lw_s\in \mathcal {C}^1(\mathbb R;L^2(\Omega ))$ and $(\rho ^{-1})_t=-2s\lambda \beta (t-\frac{T}{2}) \phi \rho ^{-1}$ in $Q_T$, we infer that

$$\begin{aligned}&\int _{Q_T} \rho ^2(t) \left( \frac{\rho ^{-2}(t) Lw_s(t)-\rho ^{-2}(t+\tau )Lw_s(t+\tau )}{\tau }\right) \\&\quad \times \left( \frac{\rho ^{-2}(t) -\rho ^{-2}(t+\tau )}{\tau }\right) Lw_s(t+\tau ) \,\text {d}x\text {d}t\\&\quad \rightarrow -2s\lambda \beta \int _{Q_T}(t-\frac{T}{2}) \phi (t) \rho ^2(t) (y_s)_ty_s\,\text {d}x\text {d}t, \end{aligned}$$

$$\begin{aligned}&\frac{1}{\tau }\int _{-\tau }^{0}\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\\&\quad \rightarrow \int _\Omega y_s(0)(\rho ^{2}y_s)_t(0)\,\text {d}x\end{aligned}$$

and

$$\begin{aligned} \frac{1}{\tau }\int _T^{T-\tau }\int _\Omega \rho ^{-2}(t+\tau )Lw_s(t+\tau )\left( \frac{L w_s(t+\tau )-Lw_s(t)}{\tau }\right) \,\text {d}x\text {d}t\rightarrow 0 \end{aligned}$$

as $\tau \rightarrow 0$.

Similarly, since $w_s\in \mathcal {C}^2(\mathbb R;L^2(\Omega ))$ and $(w_s)_{tx}(1,\cdot )\in L^2(-\delta ,T+\delta )$,

$$\begin{aligned}&\int _0^T \frac{\left( \eta ^2(t)\rho _1^{-2}(t) - \eta ^2(t+\tau )\rho _1^{-2}(t+\tau )\right) \left( (w_s)_x(1,t)+(w_s)_x(1,t+\tau ) \right) }{2\tau } \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times \left( \frac{(w_s)_x(1,t+\tau )-(w_s)_x(1,t)}{\tau }\right) \text {d}t\\&\rightarrow \int _0^T (\eta ^2\rho _1^{-2})_t(w_s)_x(1,t)(w_s)_{tx}(1,t)\text {d}t\end{aligned}$$

and

$$\begin{aligned} \int _{Q_T}B \mathcal {T}_\tau w_s \,\text {d}x\text {d}t\rightarrow \int _{Q_T}B (w_s)_{tt} \,\text {d}x\text {d}t\end{aligned}$$

as $\tau \rightarrow 0$. Since $(w_s)_{tt}(\cdot ,0)\in L^2(\Omega )$, the convergence (70) reads

$$\begin{aligned} \int _\Omega u_1 \mathcal {T}_\tau w_s(\cdot ,0) \,\text {d}x\rightarrow \int _\Omega (w_s)_{tt}(\cdot ,0)u_1\,\text {d}x. \end{aligned}$$

Similarly, since $(w_s)_t(\cdot ,0)\in H^1_0(\Omega )$, (69) reads

$$\begin{aligned}&\int _\Omega u_0 \mathcal {T}_\tau (w_s)_t(\cdot ,0) \,\text {d}x\rightarrow -2s\lambda \beta T\int _\Omega \phi (0)\rho ^2(0)u_0^2\,\text {d}x+ \int _\Omega \rho ^2(0) u_0u_1\,\text {d}x\\&\quad -\int _\Omega (u_0)_{x} (w_s)_{tx}(\cdot ,0) \,\text {d}x. \end{aligned}$$

We conclude that the limit with respect to $\tau \rightarrow 0$ in (62) leads to the following equality

$$\begin{aligned}&\int _{Q_T} \rho ^2(t)\left| (y_s)_t\right| ^2 \,\text {d}x\text {d}t+s\int _0^T \eta ^2(t)\rho _1^{-2}(t) \left| (w_s)_{tx}(1,t)\right| ^2 \text {d}t\nonumber \\&\quad =-2s\lambda \beta \int _{Q_T}(t-\frac{T}{2}) \phi (t) \rho ^2(t) (y_s)_ty_s\,\text {d}x\text {d}t- \int _\Omega y_s(0)( \rho ^2y_s)_t(0)\nonumber \\&\quad +s\int _0^T (\eta ^2\rho _1^{-2})_t(t)(w_s)_x(1,t)(w_s)_{tx}(1,t)\text {d}t\nonumber \\&\quad - \int _{Q_T}B (w_s)_{tt} \,\text {d}x\text {d}t-\int _\Omega (w_s)_{tt}(\cdot ,0)u_1\,\text {d}x\nonumber \\&\quad -2s\lambda \beta T\int _\Omega \phi (0)\rho ^2(0)u_0^2\,\text {d}x+ \int _\Omega \rho ^2(0) u_1 u_0\,\text {d}x\nonumber \\&\quad -\int _\Omega (u_0)_{x} (w_s)_{tx}(\cdot ,0) \,\text {d}x\nonumber \\&\quad :=K_1+K_2+K_3+K_4+K_5+K_6+K_7+K_8. \end{aligned}$$

(73)

Sub-step 2 : In this step, we estimate each term $K_i, i=1,\cdots ,8$.

(i) We get that, there exists $C>0$ only depending on T such that

$$|K_1|\leqslant \frac{1}{8}\int _{Q_T} \rho ^2(t) |(y_s)_t|^2\,\text {d}x\text {d}t+Cs^2\int _{Q_T} \rho ^2(t) |y_s|^2\,\text {d}x\text {d}t.$$

(ii) Similarly, recalling that $y_s(\cdot ,0)=u_0$ and $(y_s)_t(\cdot ,0)=u_1$, there exists $C>0$ such that

$$|K_2|\leqslant Cs \Vert \rho (0)u_0\Vert ^2_{L^2(\Omega )}+ \Vert \rho (0)u_0\Vert _{L^2(\Omega )}\Vert \rho (0)u_1\Vert _{L^2(\Omega )}.$$

(iii) Using that $(\rho _1^{-1})_t=-2s\lambda \beta (t-\frac{T}{2}) \phi \rho _1^{-1}$, we obtain

$$\begin{aligned}&|K_3|=\left| s\int _0^T (\eta ^2\rho _1^{-2})_t(w_s)_x(1,t)(w_s)_{tx}(1,t)\text {d}t\right| \\&\leqslant 2\left( s\int _0^T |(\eta \rho _1^{-1})_t|^2|(w_s)_x(1,t)|^2 \text {d}t\right) ^{1/2}\left( s\int _0^T \eta ^2(t)\rho _1^{-2}(t)| (w_s)_{tx}(1,t)|^2\text {d}t\right) ^{1/2}\\&\leqslant C\left( s^3\int _0^T \eta ^2(t)\rho _1^{-2}(t)|(w_s)_x(1,t)|^2 \text {d}t+s\int _0^T\rho _1^{-2}(t)|(w_s)_x(1,t)|^2 \text {d}t\right) ^{1/2}\\&\times \left( s\int _0^T \eta ^2(t)\rho _1^{-2}(t)| (w_s)_{tx}(1,t)|^2\text {d}t\right) ^{1/2}\\&\leqslant C\left( s\int _\delta ^{T-\delta } \frac{\rho _1^2(t)}{\eta ^2(t)}v_s^2 \text {d}t+s\int _0^T\rho _1^{-2}(t)|(w_s)_x(1,t)|^2 \text {d}t\right) \\&\qquad +\frac{s}{8}\int _0^T \eta ^2(t)\rho _1^{-2} | (w_s)_{tx}(1,t)|^2\text {d}t. \end{aligned}$$

We now estimate the term $\int _0^T\rho _1^{-2}|(w_s)_x(1,t)|^2 \text {d}t$ appearing in the previous inequality: proceeding as in [23, Lemma 3.7] with $q(x,t)=x\rho ^{-2}(x,t)$ such that $q(0,t)=0$ and $q(1,t)=\rho _1^{-2}(t)$, we get the equality

$$\begin{aligned}&\frac{1}{2}\int _0^T \rho _1^{-2}(t)|(w_s)_x(1,t)|^2\\&\quad =2\int _{Q_T} x \rho ^{-1}\rho _t^{-1} w_x w_t +\frac{1}{2}\int _{Q_T} (\rho ^{-2}-2x\rho ^{-1}\rho _x^{-1})(w_x^2+w_t^2)\\&\qquad +\int _\Omega [x\rho ^{-2} w_tw_x]_0^T. \end{aligned}$$

Writing that $\vert \rho ^{-1}\rho _x^{-1}\vert \leqslant Cs \rho ^{-2}$ and $\vert \rho ^{-1}\rho _t^{-1}\vert \leqslant Cs \rho ^{-2}$, we obtain (since $s\geqslant 1$)

$$\begin{aligned}&\frac{1}{2}\int _0^T \rho _1^{-2}(t)|(w_s)_x(1,t)|^2\\&\quad \leqslant Cs\int _{Q_T} \rho ^{-2}(w_x^2+w_t^2) +Cs\int _\Omega \biggl (\rho ^{-2}(0)(w_t^2+w_x^2)(0)\\&\qquad +\rho ^{-2}(T)(w_t^2+w_x^2)(T)\biggr ) \end{aligned}$$

leading, using the Carleman estimate (10), to

$$\begin{aligned} s\int _0^T\rho _1^{-2}|(w_s)_x(1,t)|^2 \text {d}t\leqslant Cs\left( \Vert \rho y_s\Vert _{L^2(Q_T)}^2+s^{-1}\Vert \frac{\rho _1}{\eta }v_s\Vert _{L^2(\delta ,T-\delta )}^2\right) . \end{aligned}$$

(74)

Thus,

$$|K_3|\leqslant Cs\left( \Vert \rho y_s\Vert _{L^2(Q_T)}^2+ \Vert \frac{\rho _1}{\eta }v_s\Vert _{L^2(\delta ,T-\delta )}^2\right) +\frac{s}{8}\int _0^T \eta ^2(t)\rho _1^{-2} (w_s)_{tx}(1,t)\text {d}t.$$

(iv) Using the Carleman estimate (72) we have

$$\begin{aligned}&|K_4|=\left| \int _{Q_T}B (w_s)_{tt} \,\text {d}x\text {d}t\right| \leqslant \left( s^{-1}\int _{Q_T}\rho ^2B^2\right) ^{1/2} \left( s\int _{Q_T}\rho ^{-2}|(w_s)_{tt}|^2\right) ^{1/2}\\&\leqslant C\left( s^{-1}\int _{Q_T}\rho ^2B^2\right) ^{1/2} \left( \int _{Q_T}\rho ^{-2} |(\rho ^2 y_s)_t |^2 \,\text {d}x\text {d}t+ s \int _0^T \eta ^2(t)\rho ^{-2}_1 |(w_s)_{tx}(1,t)|^2 \text {d}t\right) ^{1/2}\\&\leqslant C\left( s^{-1}\int _{Q_T}\rho ^2B^2\right) ^{1/2} \left( s^2 \int _{Q_T}\rho ^{2} | y_s |^2 \,\text {d}x\text {d}t+\int _{Q_T}\rho ^{2}|( y_s)_t |^2 \,\text {d}x\text {d}t\right. \\&\quad \left. + s \int _0^T \eta ^2(t)\rho ^{-2}_1 |(w_s)_{tx}(1,t)|^2 \text {d}t\right) ^{1/2}\\&\leqslant C\left( s^{-1}\int _{Q_T}\rho ^2B^2 + s^2 \int _{Q_T}\rho ^{2} | y_s |^2 \,\text {d}x\text {d}t\right) +\frac{1}{8}\int _{Q_T}\rho ^{2} |( y_s)_t |^2 \,\text {d}x\text {d}t\\&\quad + \frac{s}{8} \int _0^T \eta ^2(t)\rho ^{-2}_1 |(w_s)_{tx}(1,t)|^2 \text {d}t. \end{aligned}$$

(v) Similarly, using again the Carleman estimate (72) we have

$$\begin{aligned} |K_5|&=\left| \int _\Omega (w_s)_{tt}(\cdot ,0)u_1\,\text {d}x\right| \leqslant \Vert \rho (0)u_1\Vert _{L^2(\Omega )} \Vert \rho ^{-1}(0)(w_s)_{tt}(\cdot ,0)\Vert _{L^2(\Omega )}\\&\leqslant C s^{-1/2}\Vert \rho (0) u_1\Vert _{L^2(\Omega )} \left( \int _{Q_T}\rho ^{-2}(t) |(\rho ^2 y_s)_t |^2 \,\text {d}x\text {d}t\right. \\&\quad \left. + s \int _0^T \eta ^2(t)\rho ^{-2}_1 |(w_s)_{tx}(1,t)|^2 \text {d}t\right) ^{1/2}\\&\leqslant C s^{-1/2}\Vert \rho (0) u_1\Vert _{L^2(\Omega )} \left( s^2 \int _{Q_T}\rho ^{2} | y_s |^2 \,\text {d}x\text {d}t+\int _{Q_T}\rho ^{2} |( y_s)_t |^2 \,\text {d}x\text {d}t\right. \\&\left. + s \int _0^T \eta ^2(t)\rho ^{-2}_1 |(w_s)_{tx}(1,t)|^2 \text {d}t\right) ^{1/2}\\&\leqslant C\left( s^{-1}\Vert \rho (0) u_1\Vert ^2_{L^2(\Omega )} + s^2 \int _{Q_T}\rho ^{2} | y_s |^2 \,\text {d}x\text {d}t\right) +\frac{1}{8} \int _{Q_T}\rho ^{2} |( y_s)_t |^2 \,\text {d}x\text {d}t\\&\quad + \frac{s}{8} \int _0^T \eta ^2(t)\rho ^{-2}_1 |(w_s)_{tx}(1,t)|^2 \text {d}t. \end{aligned}$$

(vi) Simpler, we get

$$|K_6|\leqslant Cs\Vert \rho (0)u_0\Vert ^2_{L^2(\Omega )} $$

(vii) and

$$ |K_7|\leqslant \Vert \rho (0)u_0\Vert _{L^2(\Omega )}\Vert \rho (0)u_1\Vert _{L^2(\Omega )}. $$

(viii) Eventually, (72) leads to

$$\begin{aligned} |K_8|&=\left| \int _\Omega (u_0)_{x} (w_s)_{tx}(\cdot ,0)\right| \leqslant \Vert \rho (0)(u_0)_x\Vert _{L^2(\Omega )} \Vert \rho ^{-1}(0)(w_s)_{tx}(\cdot ,0)\Vert _{L^2(\Omega )}\\&\leqslant Cs^{-1/2} \Vert \rho (0) (u_0)_{x}\Vert _{L^2(\Omega )} \left( \int _{Q_T}\rho ^{-2}(t) |(\rho ^2 y_s)_t |^2 \,\text {d}x\text {d}t\right. \\&\quad \left. + C s \int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(w_s)_{tx}(1,t)|^2 \text {d}t\right) ^{1/2}\\&\leqslant Cs^{-1/2} \Vert \rho (0) (u_0)_{x}\Vert _{L^2(\Omega )} \left( s^2 \int _{Q_T}\rho ^{2}(t) | y_s |^2 \,\text {d}x\text {d}t+\int _{Q_T}\rho ^{2}(t) |( y_s)_t |^2 \,\text {d}x\text {d}t\right. \\&\quad + \left. s \int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(w_s)_{tx}(1,t)|^2 \text {d}t\right) ^{1/2}\\&\leqslant C \left( s^{-1} \Vert \rho (0) (u_0)_{x}\Vert ^2_{L^2(\Omega )} + s^2 \int _{Q_T}\rho ^{2}(t) | y_s |^2 \,\text {d}x\text {d}t\right) \\&\quad +\frac{1}{8}\int _{Q_T}\rho ^{2}(t) |( y_s)_t |^2 \,\text {d}x\text {d}t\\&\quad + \frac{s}{8} \int _0^T \eta ^2(t)\rho ^{-2}_1(t) |(w_s)_{tx}(1,t)|^2 \text {d}t. \end{aligned}$$

Sub-step 3 : In this step, we give estimates on $(v_s)_t$ and $(y_s)_t$. Collecting the previous estimates, we get from (73)

$$\begin{aligned} \begin{aligned} \int _{Q_T}&\rho ^2(t)\left| (y_s)_t\right| ^2 \,\text {d}x\text {d}t+s\int _0^T \eta ^2(t)\rho _1^{-2}(t) \left| (w_s)_{tx}(1,t)\right| ^2 \text {d}t\\&\leqslant C\left( s^2\Vert \rho y_s \Vert ^2_{L^2(Q_T)}+s \Vert \frac{\rho _1}{\eta }v\Vert _{L^2(\delta ,T-\delta )}^2 + s^{-1}\int _{Q_T}\rho ^2B^2 \right. \\&\qquad \left. +s\Vert \rho (0)u_0\Vert ^2_{L^2(\Omega )} +s^{-1}\Vert \rho (0)(u_0)_x\Vert _{L^2(\Omega )} ^2+s^{-1}\Vert \rho (0) u_1\Vert _{L^2(\Omega )}^2\right) . \end{aligned} \end{aligned}$$

(75)

We have

$$\begin{aligned} \begin{aligned}&s^{-1}\int _0^T \rho _1^2(t) \left| (v_s)_t\right| ^2dt\\&\leqslant C\left( s\int _0^T \eta ^2(t)\rho _1^{-2}(t) \left| (w_s)_{tx}(1,t)\right| ^2+s\int _\delta ^{T-\delta } \frac{\rho _1^2(t)}{\eta ^2(t)}v_s^2(t) \text {d}t\right. \\&\quad \left. +s\int _0^T \rho _1^{-2}(t) \left| (w_s)_{x}(1,t)\right| ^2\right) \end{aligned} \end{aligned}$$

(76)

thus using the estimates (74) and (14), (75) implies for $s\geqslant s_0\geqslant 1$ that

$$\begin{aligned} \begin{aligned}&\int _{Q_T} \rho ^2(t)\left| (y_s)_t\right| ^2 \,\text {d}x\text {d}t+s^{-1}\int _0^T \rho _1^2(t) \left| (v_s)_t\right| ^2dt\\&\leqslant C \left( s^{-1} \Vert \rho B\Vert ^2_{L^2(Q_T)} +se^{-2s}\Vert u_0\Vert ^2_{L^2(\Omega )} +s^{-1}e^{-2s}\Vert (u_0)_x\Vert _{L^2(\Omega )} ^2\right. \\&\quad \left. +s^{-1}e^{-2s}\Vert u_1\Vert _{L^2(\Omega )}^2\right) \\ \end{aligned} \end{aligned}$$

(77)

which gives the announced estimate (19) in the case of regular data.

Step 3 : Case where $B\in L^2(Q_T)$ and $(u_0,u_1)\in H^1_0(\Omega )\times L^2(\Omega )$. We proceed by density: there exist $(u_0^n)_{n\in \mathbb N}\in H^2(\Omega )\cap H^1_0(\Omega )$, $(u_1^n)_{n\in \mathbb N}\in H^1_0(\Omega )$ and $(B^n)_{n\in \mathbb N}\in \mathcal {D}(0,T;L^2(\Omega ))$ such that $u_0^n\rightarrow u_0$ in $H^1_0(\Omega )$, $u_1^n\rightarrow u_1$ in $L^2(\Omega )$ and $B^n\rightarrow B$ in $L^2(Q_T)$ as $n\rightarrow \infty$.

Let $(y^n_s,v^n_s)$ be the solution of (6) given in Theorem 6 associated to $(u_0^n, u_1^n,B^n)$. Then, by linearity, we have for all $(n,m)\in \mathbb N^2$, from (14)

$$\begin{aligned}&\Vert \rho (y^n_s-y^m_s)\Vert _{L^2(Q_T)} + s^{-1/2}\Vert \rho _1( v^n_s-v^m_s)\Vert _{L^2(0,T)} \\&\quad \leqslant \Vert \rho (y^n_s-y^m_s)\Vert _{L^2(Q_T)} + s^{-1/2}\left\| \frac{\rho _1}{\eta } (v^n_s-v^m_s)\right\| _{L^2(\delta ,T-\delta )}\\&\quad \leqslant C \left( s^{-3/2} \Vert \rho (B^n-B^m)\Vert _{L^2(Q_T)} + s^{-1/2} e^{-s} \Vert u_0^n-u_0^m\Vert _{L^2(\Omega )} \right. \\&\quad \left. + s^{-3/2} e^{-s} \Vert u_1^n-u_1^m\Vert _{L^{2}(\Omega )} \right) , \end{aligned}$$

while from (19)

$$\begin{aligned} \Vert \rho (y^n_s-y^m_s)_t \Vert _{L^2(Q_T)}&+ s^{-1/2} \Vert \rho _1 (v^n_s-v^m_s)_t \Vert _{L^2(0,T)} \\ \leqslant&\left( s^{-1/2} \Vert \rho (B^n-B^m)\Vert _{L^2(Q_T)} + s^{1/2} e^{-s} \Vert u_0^n-u_0^m\Vert _{L^2(\Omega )} \right. \\&\left. +s^{-1/2}e^{-s} \Vert (u_0^n-u_0^m)_x\Vert _{L^2(\Omega )} + s^{-1/2} e^{-s} \Vert u_1^n-u_1^m\Vert _{L^2(\Omega )}\right) \end{aligned}$$

and from (21).

$$\begin{aligned}&\Vert (y^n_s-y^m_s)_t \Vert _{L^\infty (0,T;L^2(\Omega ))}+\Vert (y^n_s-y^m_s)_x \Vert _{L^\infty (0,T;L^2(\Omega ))}\\&\quad \leqslant C \left( \Vert B^n-B^m\Vert _{L^2(Q_T)} +\Vert u_0^n-u_0^m\Vert _{L^2(\Omega )}\right. \\&\quad \left. + \Vert (u_1^n-u_1^m)_x\Vert _{L^2(\Omega )}+\Vert v^n_s-v^m_s \Vert _{H^1(0,T)}\right) . \end{aligned}$$

Therefore $v^n_s\rightarrow v_s$ in $H^1(0,T)$ and $y^n_s\rightarrow y_s\in \mathcal {C}^0([0,T];H^1(\Omega ))\cap \mathcal {C}^1([0,T];L^2(\Omega ))$ and, passing to the limit in the equation (6) satisfied by $(y^n,v^n)$, we obtain that $(y_s,v_s)$ solves (6). Moreover, passing to the limit in the estimate (19) satisfied by $(y^n_s,v^n_s)$, we deduce that $(y_s,v_s)$ also satisfies (19). Using (10), we easily check that $(y_s,v_s)$ satisfies $v_s = s\eta ^2\rho _1^{-2}(w_s)_x(1,\cdot )$ and $y_s=\rho ^{-2}L w_s$ where $w_s\in P_s$ is the unique solution of (12). The proof of Theorem 7 is complete.