Abstract
In this article, we consider a system consisting of two elastic strings with attached tip masses coupled through an elastic spring. Our aim is to analyze its exact boundary controllability properties and to characterize the spaces of controllable initial data depending on the number of controls acting on the boundaries of the strings. We show that singularities in waves are “smoothed by three orders” as they cross a point mass. Consequently, when only one control acts on the extremity of the first string, the space of controlled initial data is asymmetric, the components corresponding to the second string having to be more regular than those corresponding to the first one. Roughly speaking, if the initial data for the string which is directly controlled can be in \(L^2\times H^{-1}\), they should be at least in \(H^3\times H^2\) for the second string, located on the other part of the masses.
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1 Introduction
In this paper, we consider a hyperbolic system with dynamic boundary conditions, that is, with boundary operators containing time derivatives. Besides the typical Dirichlet and Neumann boundary conditions, hyperbolic equations with dynamic boundary conditions appear in modeling structures where long and slender members are linked to smaller and heavier bodies. A well-known example is the SCOLE (NASA Spacecraft Control Laboratory Experiment) model which describes the movement in one plane of a beam clamped at one end and connected to a rigid body at the other end. The importance of this model stems from it being used to model the vibrations of a flexible mast holding an antenna on a spacecraft. More recently, it is being used also to model the vibrations of the tower of a wind-turbine or of a structure consisting of a robotic arm attached to a satellite. The corresponding dynamical systems may be regarded as hybrid ones in that the elastic vibrations of the long and flexible structure are governed by a partial differential equation (for instance, the PDE of Euler–Bernoulli in linear elasticity theory), whereas the oscillations of the bodies are described by ordinary differential equations (the ODE of Newton–Euler in rigid body dynamics). Gradually, these hybrid systems became more and more complex being composed of several flexible structures joined together by different coupling mechanisms. The interested reader is referred to, for instance, [1, 4,5,6, 8, 11, 12, 14, 15, 21] and the references therein for more details on these models.
The aim of this article is to study the problem of exact boundary controllability for a hybrid system consisting of two 1-D linear wave equations with dynamic boundary conditions. More precisely, we consider two elastic strings with the same length located on the interval (0, L). At the extremity \(x=0\) each string has attached a unit mass. These masses are connected by an elastic spring (see Fig. 1). By Hooke’s Laws, we can use coupled dynamic boundary conditions to describe the dynamical behavior of this extremity. At the end \(x = L\) we act with a couple of controls \((h^1,h^2)\). We get the following coupled system of 1-D wave equations (see [19]):
where \(\kappa \) stands for the stiffness (Hooke’s constant) of the spring. In the sequel we’ll suppose that \(\kappa \) is a positive constant. We want to find boundary controls \((h^1, h^2)\) on \(x = L\) in order to obtain the exact boundary controllability for the coupled system (1.1). An even more important objective is to analyze the extent to which the same goal can be achieved with a single control.
Control problems for elastic strings with attached tip masses have been under consideration for quite some time. In [8] (see also [2, 4, 5, 10]) the case of two flexible strings connected by a point mass has been considered and controllability results have been established that reflect a smoothing property due to the presence of the point mass. More precisely, by using an explicit formula for solutions of the one-dimensional wave equation, it has been proved that solutions gain one derivative when crossing the mass. Consequently, there exist spaces of solutions with different regularity on the two sides of the mass where the system is also well-posed, without having an associated natural energy. In [8] it was also observed that the spectral gap of the corresponding differential operator vanishes, fact that, as it was proved in [4], is equivalent to the existence of an asymmetric space where the system is well-posed. As it was shown in [8] for the strings connected by a point mass, the existence of asymmetric spaces where the system is well-posed has important consequences concerning the controllability properties when we act in one extremity. Indeed, in this case, if we take the control time T large enough, the space of controlled initial data is not a usual energy space, as it happens if the point mass is not present, but an asymmetric space in the sense that its elements can be less regular on the side of the mass where the control acts directly, having a difference in the number of derivatives equal to one with respect to the elements on the other side of the mass.
More complex hybrid systems consisting of elastic strings coupled via elastic springs or viscoelastic springs with tip masses have been considered in [19] which is dedicated to control theoretical investigation. A constructive method with modular structure, which generalizes that given in [18], is used to obtain a local exact boundary controllability result for the corresponding coupled quasilinear system. The aim of the present article is to have a characterization of the controllable spaces of the corresponding linear problem and to show that the smoothing property observed in [8] still holds in this new framework consisting of strings connected by elastic springs. A careful spectral analysis of the corresponding differential operator, allows us to show that the solutions gain three derivatives when crossing the masses–spring ensemble. Consequently, when we act in one extremity, the space of controllable initial data is an asymmetric one consisting of elements which are more regular on one side, the difference in the number of derivatives being this time exactly three.
The paper is organized as follows. In Sect. 2, we introduce the adjoint differential operator corresponding to our system and some spaces which will be useful in the sequel. Section 3 is devoted to the spectral analysis of the adjoint operator. All this information is used in Sect. 4 to study the controllability properties of (1.1) and to give a Fourier characterization of the controllable spaces of initial data when one or two controls are considered. The main results of the paper are given in Sect. 4 where the controllable spaces are expressed in terms of Sobolev spaces and the smoothing effect due to the dynamic boundary conditions is highlighted. The article ends with a section of comments in which several limit cases are studied and “Appendix” containing several Riesz basis properties of the trigonometric functions.
2 The adjoint system
Let us introduce some notations which will be used throughout the entire paper. Firstly, we define the space
with the usual gradient norm and the following Hilbert product space
endowed with the inner product
In (2.1)–(2.2) and in the sequel, we use \('\) to denote the transposed vector.
Secondly, we define a new Hilbert product space
with the following inner product
Notice that, due to the relations \(\varphi ^i_j(0)=\zeta ^i_j\), \(i,j\in \{1,2\}\), (2.4) does define an inner product which is equivalent to the canonical one in the product space X.
Let us introduce the following adjoint system to (1.1):
which we write in the abstract form
where \(Z= \left( \begin{array}{cccccccc} \varphi ^1&\psi ^1&\zeta ^1&\xi ^1&\varphi ^2&\psi ^2&\zeta ^2&\xi ^2\end{array} \right) '\), \(Z_0= \left( \begin{array}{cccccccc} \varphi ^1_0&\psi ^1_0&\zeta ^1_0&\xi ^1_0&\varphi ^2_0&\varphi ^2_0&\zeta ^2_0&\xi ^2_0\end{array} \right) '\) and the unbounded operator (D(A), A) in the Hilbert space X is given by
Notice that (D(A), A) is a skew-adjoint operator which generates a group of isometries \(\left( {\mathbb {T}}(t)\right) _{t\in {\mathbb {C}}}\) in X. Consequently, for any \(Z_0\in X\), there exists a unique weak solution \(Z\in C\left( {\mathbb {R}};X\right) \) of (2.5) given by \(Z(t)={\mathbb {T}}(t)Z_0\). The energy corresponding to (2.5) is defined by
and it is conserved along trajectories, i. e.
for any weak solution of (2.5).
System of two coupled strings with bodies and spring
Let \(\delta _0\) be the Dirac delta function, defined as an element in \(\left( H^1_L(0,L)\right) ^*\), the topological dual space of \(H^1_L(0,L)\):
An important role in the remaining part of this paper is played by the space \(X^*\), the topological dual of X, which will be discussed below. Firstly, let us introduce the space
endowed with the canonical inner product. We identify the space Y with its topological dual \(Y^*\) but we define a convenient duality product as follows:
for any \(\left( \begin{array}{cccccccc} \varphi ^1_j&\psi ^1_j&\zeta ^1_j&\xi ^1_j&\varphi ^2_j&\psi _j^2&\zeta ^2_j&\xi ^2_j\end{array} \right) '\in Y ,\) \(j\in \{1,2\}\). Notice that the signs and the variables coupling in the duality product (2.12) are different from the canonical inner product in Y. However, this is a valid definition of the duality product and fits our resulting moment problem in Lemma 4.1.
Now, let \({\widetilde{Y}}_{\frac{1}{2}}\) be the space
endowed with the inner product
Now, we introduce the dual space \({\widetilde{Y}}_{\frac{1}{2}}^*\) of \({\widetilde{Y}}_{\frac{1}{2}}\) with respect to the pivot space \(L^2(0,L)\times {\mathbb {C}}\). Let \(\left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\) be the dual of the product space \(H^1_L(0,L)\times {\mathbb {C}}\). According to Hahn–Banach theorem, for each element \( \Psi \in {\widetilde{Y}}_{\frac{1}{2}}^*\) there exists an extension \(\left( \begin{array}{cc}{\widehat{\psi }}&{\widehat{\alpha }} \end{array} \right) ' \in \left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\) such that
In (2.15) and in the sequel, we denote by \(\langle \,\cdot \, , \,\cdot \,\rangle _{{{\mathcal {Y}}}^*,{{\mathcal {Y}}}}\) the duality between a space \({{\mathcal {Y}}}\) and its topological dual \({{\mathcal {Y}}}^*\). From the definition of the norm of an element in a dual space and (2.14), we deduce that
Notice that the last norm does not depend on the choice of the element \(\left( \begin{array}{cc}{\widehat{\psi }}&{\widehat{\alpha }} \end{array} \right) ' \in \left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\) verifying (2.15)–(2.16). For the sake of simplicity we identify, for the remaining part of this work, an element \(\Psi \in {\widetilde{Y}}_{\frac{1}{2}}^*\) with an extension of it \(\left( \begin{array}{cc}{\widehat{\psi }}&{\widehat{\alpha }} \end{array} \right) ' \in \left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\) verifying (2.15)–(2.16).
Remark 2.1
According to a classical result (see, for instance, [16, Theorem 4.9]), the dual space \({\widetilde{Y}}_{\frac{1}{2}}^*\) of \({\widetilde{Y}}_{\frac{1}{2}}\) with respect to the pivot space \(L^2(0,L)\times {\mathbb {C}}\) is isometrically isomorphic with the quotient space of \(\left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\) with respect to the annihilator of \({\widetilde{Y}}_{\frac{1}{2}}\) given by
Denoting by \(\widehat{ \left( \begin{array}{cc} \psi&\alpha \end{array} \right) '}\) the equivalence class of \(\left( \begin{array}{cc} \psi&\alpha \end{array} \right) ' \in \left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\), this quotient space is endowed with the norm
The above considerations show that an element \(\Psi \in {\widetilde{Y}}_{\frac{1}{2}}^*\) may be identified with the class of equivalence \(\widehat{ \left( \begin{array}{cc} {\widehat{\psi }}&{\widehat{\alpha }} \end{array} \right) '}\), where \(\left( \begin{array}{cc}{\widehat{\psi }}&{\widehat{\alpha }} \end{array} \right) ' \in \left( H^1_L(0,L)\right) ^*\times {\mathbb {C}}\) verifies (2.15).
Now, we can introduce the dual space \(X^*\) of X with respect to the pivot space Y:
and we define the duality product between X and \(X^*\) as follows:
where \(\left( \begin{array}{cccccccc} \varphi ^1&\psi ^1&\zeta ^1&\xi ^1&\varphi ^2&\psi ^2&\zeta ^2&\xi ^2\end{array} \right) ' \in X\) and \(\left( \begin{array}{cccccccc} u^1&v^1&p^1&q^1&u^2&v^2&p^2&q^2\end{array} \right) '\in X^*.\)
Notice that, for uniformity, we keep the same order of components for the elements in the spaces X and \(X^*\). Thus, \(\left( \begin{array}{cccccccc} u^1&v^1&p^1&q^1&u^2&v^2&p^2&q^2\end{array} \right) '\in X^*\) means, by convention, that
In the following proposition, we introduce an inner product that makes \(X^*\) a Hilbert space.
Proposition 2.1
The operator \({{\mathcal {A}}}: X\rightarrow X^*\) defined by
is an isometric isomorphism. The map \(\langle \,\cdot \, , \,\cdot \,\rangle _{X^*}: X^*\times X^* \rightarrow {\mathbb {C}}\),
defines an inner product which introduces a structure of Hilbert space on \(X^*\).
Proof
We have that
Since we have that
and
we deduce that \(\Vert {{\mathcal {A}}}\Phi \Vert _{{\mathcal L}(X,{\mathbb {C}})}=\Vert \Phi \Vert _X\) which implies that \({\mathcal {A}}\) is an isometry.
Since \({\mathcal {A}}\) is obviously linear, in order to show that it is an isomorphism it remains to prove that it is onto. Indeed, for each \(U \in X^*\), according to Riesz representation theorem, it follows that there exists \(\Phi \in X\) such that
which gives that \({{\mathcal {A}}} \Phi =U\) and proves that \({\mathcal {A}}\) is onto.
The fact that \(\langle \,\cdot \, , \,\cdot \,\rangle _{X^*} \) defines an inner product which introduces a structure of Hilbert space on \(X^*\) is a consequence of the analogous properties of the inner product \(\langle \,\cdot \, , \,\cdot \,\rangle _X \) in X. \(\square \)
The following property shows that the operator \({{\mathcal {A}}}\) introduced in Proposition 2.1 is an extension of the operator (D(A), A) defined by (2.8).
Proposition 2.2
For each \(\Phi \in D(A)\), we have that
Proof
Both terms in (2.22) should be regarded as elements in \(X^*\) and the equality means that
The above relation follows immediately from definitions (2.8) and (2.20) of the operators (D(A), A) and \({\mathcal {A}}\), respectively. \(\square \)
3 Spectral analysis
This section is devoted to the analysis of the spectral properties of the operator (D(A), A) introduced in the previous section.
Proposition 3.1
The spectrum of the operator (D(A), A) given by (2.8) consists of two disjoint families of distinct eigenvalues
where \(\lambda _k^j=i\, \mu _{k}^j\) with \( \mu _{k}^j\in {\mathbb {R}}^*\), \(j\in \{1,2\}\), \(k\in {\mathbb {Z}}^*\). Moreover, we have that, for each \(k\in {{\mathbb {N}}^*}\),
and the following asymptotic formulas hold, with respect to the parameter k,
The eigenvectors of the operator (D(A), A) are as follows
where
Moreover, the family \((\Phi _k^1)_{k\in {\mathbb {Z}}^*} \cup (\Phi _k^2)_{k\in {\mathbb {Z}}^*}\) forms an orthonormal basis in X.
Proof
Let us consider \(\Phi =(\varphi ^1, \psi ^1, \zeta ^1, \xi ^1, \varphi ^2, \psi ^2, \zeta ^2, \xi ^2)'\) the eigenvector satisfying \(A \Phi =\lambda \Phi ,\, \lambda \in {\mathbb {C}}\). We get the following system of equations
Taking into account that \( \varphi ^1(L)= \varphi ^2(L)=0\), we obtain
where \(C,\, {\tilde{C}}\) are two constants depending on \(\lambda \) and L.
Using the above form of the solutions in the last two equations of system (3.6), we get
By adding and subtracting the last two equations, the above system becomes equivalent to
On the other hand, since the operator (D(A), A) is skew-adjoint, the eigenvalues are of the form \(\lambda = i \mu ,\) where \(\mu \in {\mathbb {R}}^*\). Finally, we have the following two cases:
a) If \(C=-{\tilde{C}}\) the values \(\mu \) are the roots of the equation
Equation (3.7) has a family of simple roots \((\mu ^1_k)_{k\in {\mathbb {Z}}^*}\) such that
In order to study the asymptotic behavior of the family \((\mu _k^1)_{k\ge 1}\), we are looking for a real constants \(\alpha \), \(\beta \) and \(\gamma \) such that
Using the above asymptotic relation in (3.7), we get (3.3).
b) If \(C={\tilde{C}}\), the values \(\mu \) are given by the roots of the equation
Equation (3.8) has a family of simple roots \((\mu ^2_k)_{k\in {{\mathbb {N}}^*}}\) such that
Moreover, based on [7, Sect. 2.3] we get (3.4). Hence, the spectrum of the operator A is given by two disjoint families of distinct eigenvalues. Moreover, since (D(A), A) is skew-adjoint, the set of the corresponding eigenvectors, given by (3.5), forms an orthonormal basis in the space X. \(\square \)
Remark 3.1
The spaces X can be characterized by using the orthonormal basis \((\Phi _k^1)_{k\in {\mathbb {Z}}^*} \cup (\Phi _k^2)_{k\in {\mathbb {Z}}^*}\). Indeed, we have that
As in the previous remark, we can describe the dual space \(X^*\) by using the set of eigenvectors of the operator (D(A), A).
Proposition 3.2
The set \(\left( \lambda _k^1 \Phi _k^1\right) _{k\in {\mathbb {Z}}^*} \cup \left( \lambda _k^2 \Phi _k^2\right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in the Hilbert space \(\left( X^*,\langle \, , \,\rangle _{X^*}\right) \) which can be characterized as follows
For any two elements \(U=\sum _{k\in {\mathbb {Z}}^*} \left( a_k^1\Phi _k^1+ a_k^2\Phi _k^2\right) \in X^*\) and \(Z=\sum _{k\in {\mathbb {Z}}^*} \left( b_k^1\Phi _k^1+ b_k^2\Phi _k^2\right) \in X\), the duality product is given by
Proof
Since the family \((\Phi _k^1)_{k\in {\mathbb {Z}}^*} \cup (\Phi _k^2)_{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in X and the operator \({{\mathcal {A}}}:X\rightarrow X^*\) defined in Proposition 2.1 is an isometric isomorphism of Hilbert spaces, it follows that \(({{\mathcal {A}}}\Phi _k^1)_{k\in {\mathbb {Z}}^*} \cup ({{\mathcal {A}}}\Phi _k^2)_{k\in {\mathbb {Z}}^*}\) forms an orthonormal basis in \(X^*\). According to Proposition 2.2, it follows that \(\left( \lambda _k^1 \Phi _k^1\right) _{k\in {\mathbb {Z}}^*} \cup \left( \lambda _k^2 \Phi _k^2\right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in \(X^*\). On the other hand, by using (2.22) we have that
from which (3.11) follows immediately. \(\square \)
4 Controllability results
The above spectral analysis enables us to translate the original control problem (1.1) into an equivalent moment problem.
Lemma 4.1
Equation (1.1) with the initial data \(U_0=\left( \begin{array}{cccccccc} u^1_0&v^1_0&p^1_0&q^1_0&u^2_0&v^2_0&p^2_0&q^2_0\end{array} \right) '\in X^*\) is null-controllable in time T if and only if there exist two controls \((h^1,h^2)\in \left[ L^2(0,T)\right] ^2\) such that
for any \(Z_0\in X\) and \(Z= \left( \begin{array}{cccccccc} \varphi ^1&\psi ^1&\zeta ^1&\xi ^1&\varphi ^2&\psi ^2&\zeta ^2&\xi ^2\end{array} \right) '\) solution of the adjoint system (2.6).
Moreover, if \(U_0=\sum _{k\in {\mathbb {Z}}^*} \left( a_k^1\Phi _k^1+a_k^2\Phi _k^2\right) \), then (4.1) is verified if and only if
Proof
Supposing that all the solutions are smooth, by multiplying (1.1) with the conjugate solution of (2.5) and integrating by parts we obtain that \(U_0\) is null-controllable in time T if and only if there exist two controls \((h^1,h^2)\in \left[ L^2(0,T)\right] ^2\) such that
which implies that (4.1) holds. Then, the moment problem (4.2) follows from (4.1) by considering \(Z_0=\Phi _k^s\), \(k\in {\mathbb {Z}}^*,\,\,s\in \{1,2\}\), and taking into account relations (3.11). \(\square \)
Now, we have all the ingredients needed to study the controllability problem (1.1). Firstly, we address the case of two controls, one for each vibrating string.
Theorem 4.1
Let \(T>2L\). The following two properties are equivalent
-
1.
The initial data \(U_0=\sum _{k\in {\mathbb {Z}}^*} \left( a_k^1\Phi _k^1+a_k^2\Phi _k^2\right) \) of system (1.1) is controllable to zero in time T by means of two controls \(h^1,h^2\in L^2(0,T)\);
-
2.
The Fourier coefficients of \(U_0\) verify
$$\begin{aligned} \sum _{k\in {\mathbb {Z}}^*} \left( \frac{|a_k^1|^2}{|\lambda _k^1|^2}+ \frac{|a_k^2|^2}{|\lambda _k^2|^2}\right) <\infty , \end{aligned}$$(4.3)i. e., according to Proposition 3.2, \(U_0\in X^*\).
Proof
From Lemma 4.1, it follows that \(U_0\) is controllable to zero in time T by means of two controls if and only if there exists \((h^1,h^2)\in \left[ L^2(0,T)\right] ^2\) such that
and
By taking into account that the asymptotic gap of the families \((\lambda _k^1)_{k\in {\mathbb {Z}}^*}\) and \((\lambda _k^2)_{k\in {\mathbb {Z}}^*}\) is, according to (3.3)–(3.4), \(\frac{\pi }{L}\), it follows that the following inequalities hold for any \(T>2L\) and any finite sequence \((\beta _k^j)_{k\in {\mathbb {Z}}}\subset {\mathbb {C}}\) (see [3]):
where \(C_j^1\) and \(C_j^2\) are positive constants.
The left parts of these inequalities imply that each of the problems of moments (4.4) and (4.5) has a solution if \(\left( \frac{i\, \alpha _k^j}{(\mu _{k}^j)^2\cos (\mu _{k}^j L)}\overline{a_k^j}\right) _{k\in {\mathbb {Z}}^*}\in \ell ^2\) for \(j=1,2\) (see, for instance, [20, Theorem 3, p. 155]). By taking into account the asymptotic formulas (3.3)–(3.4) and the expressions of \(\alpha _k^j\) from Proposition 3.1, we obtain that (4.3) implies the controllability of the initial data \(U_0\).
Reciprocally, if each of the problems of moments (4.4) and (4.5) has a solution, from the right parts of inequalities (4.6), [20, Theorem 2, p. 151]) and Banach–Steinhaus theorem we obtain that (4.3) holds and the proof ends. \(\square \)
Remark 4.1
The controllability result given by Theorem 4.1 provides the maximal space of controllable initial data \(X^*\) and also the optimal time controllability \(T>2L\). Indeed, [20, Ch. 4, Sec. 3] shows that \(T\ge 2L\) is a necessary condition for the left-hand part of inequalities (4.6).
We pass to study the controllability problem (1.1) with one control only. Without loss of generality, we suppose that \(h^2=0\) and we look for a control \(h^1\in L^2(0,T)\) in (1.1). Hence, our control is localized on the extremity \(x=L\) of the first string. Firstly, we recall the following result which is a particular case of [17, Corollary of Theorem 3]:
Theorem 4.2
Suppose that \((\omega _{k}^j)_{k\in {\mathbb {Z}},j\in \{1,2\}}\) is a set of distinct complex numbers such that
Then, the system
has a solution \(f\in L^2(-2\pi ,2\pi )\) if and only if
Moreover, if (4.8) has a solution, it is unique.
Concerning the controllability of (1.1) with one control, we have the following result.
Theorem 4.3
Let \(T\ge 4L\). The following two properties are equivalent
-
1.
The initial data \(U_0=\sum _{k\in {\mathbb {Z}}^*} \left( a_k^1\Phi _k^1+a_k^2\Phi _k^2\right) \) of system (1.1) is controllable to zero in time T by means of one control only, \(h^1\in L^2(0,T)\);
-
2.
The Fourier coefficients of \(U_0\) verify
$$\begin{aligned} \sum _{k\in {\mathbb {Z}}^*} \left( \frac{|a_k^1|^2}{|\lambda _k^1|^2}+ \frac{|a_k^1-a_k^2|^2}{|\lambda _k^2|^2|\lambda _k^1-\lambda _k^2 |^2}\right) <\infty . \end{aligned}$$(4.10)
Proof
From Lemma 4.1, it follows that \(U_0\) is controllable to zero in time T by means of one control if and only if there exist \(h^1\in L^2(0,T)\) such that
which is equivalent to
By taking into account Proposition 3.1, for each \(s\in \{0,1\}\), we have that
and
From Theorem 4.2, we obtain the desired result for \(T=4L\). Evidently, if \(T>4L\), we take \(h^1(t)=0\) for \(t\in (4L,T)\) in (4.11) and we solve the corresponding the moment problem in (0, 4L). \(\square \)
Remark 4.2
Theorem 4.3 provides the maximal space of controllable initial data which is characterized by (4.10). In the following section, we’ll describe it in terms of Sobolev spaces. Theorem 4.3 also gives the optimal time controllability \(T=4L\). On the other hand, since our family of exponential functions \(\left( e^{\lambda _k^s t}\right) _{k\in {\mathbb {Z}}^*,\, s\in \{1,2\}}\) is not complete in \(L^2(-2\pi ,2\pi )\), the corresponding moment problem (4.11) does not have a unique solution as in Theorem 4.2 even in the case \(T=4L\).
5 Asymmetric spaces
The aim of this section is to have a better understanding of the structure of the space of controllable initial data described in terms of Fourier coefficients by relation (4.10) from Theorem 4.3. More precisely, we prove that the regularity of the initial data corresponding to the second string should be higher than the one of the first string in order to be controllable from the right extremity of the latter (see Theorem 5.2 and Remark 5.5).
5.1 Some simpler spaces
Let \({\widetilde{X}}\) be the space given by (2.1) endowed with the inner product
We remark that the inner product (5.1) is equivalent to the one introduced in (2.2) and \({\widetilde{X}}\) is a Hilbert space with respect to both inner products.
Now, we consider the following auxiliary hybrid problem coupling a vibrating string located in the interval (0, L) with a mass attached at its extremity \(x=0\):
This model has been studied, for instance, in [13] where it has been shown that the corresponding differential operator is skew-adjoint and has a sequence of orthonormal eigenvectors in \({\widetilde{X}}\) with respect to the inner product (5.1) given by
where \({\widetilde{\alpha }}_k= \sqrt{ L ({\widetilde{\mu }}_k)^2+ ({\widetilde{\mu }}_k)^2\sin ^2 ({\widetilde{\mu }}_k L)+\kappa \sin ^2 ({\widetilde{\mu }}_k L)}\) and \(\left( {\widetilde{\mu }}_k\right) _{k\in {\mathbb {Z}}^*}\) are the roots of the equation
Equation (5.4) has a family of simple roots \(({\widetilde{\mu }}_k)_{k\in {\mathbb {Z}}^*}\) such that
For each \(k\ge 1\), we denote \({\widetilde{\mu }}_{-k}=-{\widetilde{\mu }}_{k}\) and, for each \(j\in \{1,2\}\) and \(k\in {\mathbb {Z}}^*\), \({{\widetilde{\lambda }}}_k=i\, {\widetilde{\mu }}_{k}\).
Moreover, for each \(\gamma \ge 0\), let us define the space
We denote by \(\Vert \,\cdot \Vert _{{\widetilde{X}}_\gamma }\) the corresponding norm in \({\widetilde{X}}_\gamma \).
Remark 5.1
If \(\left( \lambda _k^s\right) _{k\in {\mathbb {Z}}^*,\, s\in \{1,2\}}\) are the eigenvalues given in Proposition 3.1 and \(\delta _k=|\lambda _k^1-\lambda _k^2|\), we have that the space \({\widetilde{X}}_\gamma \) can be equivalently characterized by
Notice that, in the case \(\gamma =2\), the weight \(|\lambda _k^1|^2\delta _k^{\frac{2(\gamma +1)}{3}}\) appearing in the above characterization of the space \({\widetilde{X}}_\gamma \) coincides with the one from (4.10) in Theorem 4.3 which gives the controllable initial data.
Remark 5.2
Since the differential operator corresponding to (5.2) is skew-adjoint, \(\left( {\widetilde{\Phi }}_k \right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in \({\widetilde{X}}\) with respect to the inner product (2.4). Hence, for any \(U\in {\widetilde{X}}\), there exists a unique sequence \((b_k)_{k\in {\mathbb {Z}}^*}\in \ell ^2\) such that
Let us show how can each component of \(U=(U_1,U_2,U_3,U_4)'\) be written in the Riesz basis given by Lemma 7.2 in “Appendix.” From (5.7), we deduce that:
-
\(\displaystyle U_1=\sum _{k\ge 1}\frac{{\tilde{\mu }}_k}{{\widetilde{\alpha }}_k}(b_k-b_{-k})\frac{1}{{\tilde{\mu }}_k}\sin ({\tilde{\mu }}_k(x-L)).\)
-
\(\displaystyle U_2=\sum _{k\ge 2} \left( -\frac{{\tilde{\mu }}_k\, i}{{\widetilde{\alpha }}_k}(b_k+b_{-k})-\frac{{\tilde{\mu }}_1\, i}{{\widetilde{\alpha }}_1}(b_1+b_{-1})c_k \right) \sin ({\tilde{\mu }}_k(x-L)),\) where \(c_k\) are given by the expansion \(\sin ({\tilde{\mu }}_1(x-L))=\sum _{k\ge 2} c_k \sin ({\tilde{\mu }}_k(x-L)).\)
-
The values \(U_3\in {\mathbb {C}}\) is given by \(U_1(0)\).
-
\(\displaystyle U_4=\sum _{k\ge 1} \frac{{\tilde{\mu }}_k\, i}{{\widetilde{\alpha }}_k}(b_k+b_{-k})\sin ({\tilde{\mu }}_kL).\)
We remark that, given \((a_k^1)_{k\ge 1},\,\, (a_k^2)_{k\ge 2}\in \ell ^2\) and \(U_4\in {\mathbb {C}}\), there exists a unique sequence \((b_k)_{k\in {\mathbb {Z}}^*}\in \ell ^2\) such that:
Indeed, the above system can be equivalently written as follows:
Thus, for each element \(U\in {\widetilde{X}}\), we have the following two equivalent expansions
which are related through relations (5.8).
The following result is a consequence of Lemma 7.2 from “Appendix” concerning the Riesz basis properties of the trigonometric functions introduced there. The main issue is the characterization of some of the spaces \({\widetilde{X}}_\gamma \) in terms of Sobolev spaces.
Proposition 5.1
For each \(\gamma \ge 0\), let \({\widetilde{X}}_\gamma \) be the space given by (5.6). We have the following properties:
-
1.
\({\widetilde{X}}_{0}={\widetilde{X}}\).
-
2.
The space \({\widetilde{X}}_2\) is given by
$$\begin{aligned} {\widetilde{X}}_2=\displaystyle \left\{ \left( \begin{array}{cccccccc} \varphi&\psi&\zeta&\xi \end{array} \right) ' \in H^3(0,L) \times H^2(0,L)\times {\mathbb {C}}\times {\mathbb {C}} \,:\, \right. \\ \left. \varphi (L)=\varphi _{xx}(L)=\psi (L)=0,\,\, \varphi (0)=\zeta ,\,\, \psi (0)=\xi \right\} .\end{aligned}$$
Proof
-
1.
It follows immediately from the fact that \(\left( {\widetilde{\Phi }}_k\right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in \({\widetilde{X}}\).
-
2.
We denote
$$\begin{aligned}V=\displaystyle \left\{ \left( \begin{array}{cccccccc} \varphi&\psi&\zeta&\xi \end{array} \right) ' \in H^3(0,L) \times H^2(0,L)\times {\mathbb {C}}\times {\mathbb {C}} \right. \,:\, \\ \left. \varphi (L)=\varphi _{xx}(L)=\psi (L)=0,\,\, \varphi (0)=\zeta ,\,\, \psi (0)=\xi \right\} ,\end{aligned}$$and we prove 2. by double inclusion method. If \(U=(U_1,U_2,U_3,U_4)'\in {\widetilde{X}}_2\), then there exists \(\left( b_k\right) _{k\in {\mathbb {Z}}^*}\) such that \(\sum _{k\in {\mathbb {Z}}^*} |b_k|^2|{\widetilde{\mu }}_k|^4<\infty \) and
$$\begin{aligned} U= \sum _{k\in {\mathbb {Z}}^*} b_k{\widetilde{\Phi }}_k. \end{aligned}$$As in Remark 5.2, it follows that
-
\(\displaystyle U_1=\sum _{k\ge 1}\frac{b_k-b_{-k}}{{\widetilde{\alpha }}_k}\sin ({\tilde{\mu }}_k(x-L)).\)
-
\(\displaystyle U_2=\sum _{k\ge 1} -\frac{{\tilde{\mu }}_k\, i\, (b_k+b_{-k})}{{\widetilde{\alpha }}_k} \sin ({\tilde{\mu }}_k(x-L)),\)
-
\(U_3=\displaystyle \sum _{k\ge 1}-\frac{b_k-b_{-k}}{{\widetilde{\alpha }}_k}\sin ({\tilde{\mu }}_k L)=U_1(0)\).
-
\(\displaystyle U_4=\sum _{k\ge 1} \frac{{\tilde{\mu }}_k\, i}{{\widetilde{\alpha }}_k}(b_k+b_{-k})\sin ({\tilde{\mu }}_kL)=U_2(0).\)
Since \({\widetilde{\alpha }}_k\sim {\widetilde{\mu }}_k\) as k tends to infinity, it follows from the above expressions, the properties of the sequence \(\left( b_k\right) _{k\in {\mathbb {Z}}^*}\) and the characterization of the spaces \({{\mathcal {X}}}_2\) and \({{\mathcal {X}}}_3\) in Lemma 7.2 that \(U\in V\). Reciprocally, if \(U=(U_1,U_2,U_3,U_4)'\in V\), then it follows from Lemma 7.2 that there exists two sequences \(\left( a_k^1\right) _{k\ge 1}\) and \(\left( a_k^2\right) _{k\ge 1}\) such that
-
\(\displaystyle U_1=\sum _{k\ge 1}a_k^1\sin ({\tilde{\mu }}_k(x-L)) \hbox { with } \sum _{k\ge 1} |a_n^1|^2|{\tilde{\mu }}_k|^{6}<\infty ,\)
-
\(\displaystyle U_2=\sum _{k\ge 1}a_k^2 \sin ({\tilde{\mu }}_k(x-L)) \hbox { with } \sum _{k\ge 1} |a_n^2|^2|{\tilde{\mu }}_k|^{4}<\infty ,\)
-
\(U_3=\displaystyle \sum _{k\ge 1}-a_k^1 \sin ({\tilde{\mu }}_k L)=U_1(0)\),
-
\(\displaystyle U_4=\sum _{k\ge 1}-a_k^2 \sin ({\tilde{\mu }}_kL)=U_2(0).\)
From Remark 5.2, we deduce that there exists a sequence \(\left( b_k\right) _{k\in {\mathbb {Z}}^*}\subset {\mathbb {C}}\) with the property that \(U= \sum _{k\in {\mathbb {Z}}^*} b_k{\widetilde{\Phi }}_k,\) where
$$\begin{aligned} b_{\pm 1}=\pm \frac{{\widetilde{\alpha }}_1}{2{\tilde{\mu }}_1}a_1^1 ,\quad b_{\pm k}=\frac{{\widetilde{\alpha }}_k}{2{\tilde{\mu }}_k}\left( \pm a_k^1+i a_k^2 \right) \qquad (k\ge 2). \end{aligned}$$Thus, \(\sum _{k\in {\mathbb {Z}}^*} |b_k|^2 |{\widetilde{\mu }}_k|^4<\infty \), which implies that \(U\in {\widetilde{X}}_2\) and the proof ends.
-
\(\square \)
We can describe the dual space \({\widetilde{X}}^*\) of \({\widetilde{X}}\) by using the set of eigenvectors \(\left( {\widetilde{\Phi }}_k \right) _{k\in {\mathbb {Z}}^*}\). As in Proposition 2.1, we can define an inner product \(\langle \,\cdot \,, \,\cdot \,\rangle _{{\widetilde{X}}^*}\) which induces a Hilbert space structure on \({\widetilde{X}}^*\). The following result is similar to Proposition 3.2 and we omit its proof.
Proposition 5.2
The space \({\widetilde{X}}^*\) is isomorphic with the space \(\displaystyle L^2(0,L)\times {\mathbb {C}}\times {\widetilde{Y}}_{\frac{1}{2}}^*\), where \({\widetilde{Y}}_{\frac{1}{2}}^*\) is the dual space of \({\widetilde{Y}}_{\frac{1}{2}}\) given by (2.13), with respect to the pivot space \(L^2(0,L)\times {\mathbb {C}}\). Moreover, \(\left( {\widetilde{\mu }}_k {\widetilde{\Phi }}_k\right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in the Hilbert space \(\left( {\widetilde{X}}^*,\langle \, , \,\rangle _{{\widetilde{X}}^*}\right) \) which can be characterized as follows
If \(D(({\widetilde{A}}),{\widetilde{A}})\) is the differential operator corresponding to (5.2),
and, for any \(\Phi \in D({\widetilde{A}})\), we have that
Remark 5.3
Taking into account the characterization (3.10) of the dual space \({\widetilde{X}}^*\) and definition (5.6) of the spaces \({\widetilde{X}}_\gamma \), in the sequel we shall use the following notation:
5.2 Two new bases in the simpler spaces
Let us write the eigenvectors of the operator (D(A), A) from Sect. 3 as follows:
where
We can immediately see that \({\widehat{\Phi }}_k^j\in {\widetilde{X}}\) for each \(k \in {{\mathbb {Z}}^*}\) and \(j\in \{1,2\}\). However, much more can be said concerning the sets of vectors \(\left( {\widehat{\Phi }}_k^j\right) _{k\in {\mathbb {Z}}^*}\) and we have the following results.
Proposition 5.3
For each \(j\in \{1,2\}\), the set \(\left( {\widehat{\Phi }}_k^j\right) _{k\in {\mathbb {Z}}^*}\) is a Riesz basis in \({\widetilde{X}}\).
Proof
Let \(j\in \{1,2\}\). Firstly, we remark that \(\left( {\widehat{\Phi }}_k^j\right) _{k\in {\mathbb {Z}}^*}\) is an \(\omega \)-linearly independent set in \({\widetilde{X}}\). Indeed, from the fact that
it follows that
and, by taking into account that the family \((\Phi _k^1)_{k\in {\mathbb {Z}}^*} \cup (\Phi _k^2)_{k\in {\mathbb {Z}}^*}\) forms an orthonormal basis in X, we deduce that \(a_k=0\) for each \(k\in {\mathbb {Z}}^*.\) On the other hand, we recall that \(\left( {\widetilde{\Phi }}_k\right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in \({\widetilde{X}}\). Moreover, by taking into account the asymptotic relations (3.3), (3.4) and (5.5), we have that
for some positive constant C. From the above relations, we deduce that
and, from [20, Theorem 15, Ch. 1], it follows that \(\left( {\widehat{\Phi }}_k^j\right) _{k\in {\mathbb {Z}}^*}\) is a Riesz basis in \({\widetilde{X}}\). \(\square \)
In fact, we can extend the previous result to some other spaces \({\widetilde{X}}_\gamma \).
Proposition 5.4
Let \(\gamma \in \left\{ -1, 2\right\} \). The families \(\left( (\lambda _k^1)^{-\gamma }\; {\widehat{\Phi }}_k^1\right) _{k\in {\mathbb {Z}}^*}\), \(\left( (\lambda _k^1)^{-\gamma }\; {\widehat{\Phi }}_k^2\right) _{k\in {\mathbb {Z}}^*}\) and \(\left( ({\widetilde{\lambda }}_{k})^{-\gamma } \; {\widetilde{\Phi }}_k\right) _{k\in {\mathbb {Z}}^*}\) are equivalent basis in \({\widetilde{X}}_\gamma \).
Proof
From the definition of the spaces \({\widetilde{X}}_{\gamma }\) and Propositions 5.1–5.2, we deduce that \(\left( ({\widetilde{\lambda }}_{k})^{-\gamma } \; {\widetilde{\Phi }}_k\right) _{k\in {\mathbb {Z}}^*}\) is an orthonormal basis in \({\widetilde{X}}_\gamma \). For each \(j\in \{1,2\}\) and \(\gamma \in \left\{ -1,2\right\} \) we show that
which, by using a similar argument as in Proposition 5.3, allows us to conclude the proof.
In order to show (5.17), it suffices to prove the following estimate
Firstly, we remark that
and
If \(\gamma =2\), (5.18) follows from (5.19)–(5.20) and the characterization of the space \({\widetilde{X}}_{2}\) from Proposition 5.1 which allows to prove that
If \(\gamma =-1\), from (5.11) in Proposition 5.2 we obtain that
and therefore
Since the asymptotic relation (5.5) implies that \(\left| \lambda _k^1-{\widetilde{\lambda }}_{k}\right| ={\mathcal O}\left( \frac{1}{k^3}\right) \), it follows that (5.18) holds for \(\gamma \in \{-1,2\}\) and the proof is complete. \(\square \)
Remark 5.4
Since the families \(\left( (\lambda _k^1)^{-\gamma }\; {\widehat{\Phi }}_k^1\right) _{k\in {\mathbb {Z}}^*}\), \(\left( (\lambda _k^1)^{-\gamma }\; {\widehat{\Phi }}_k^2\right) _{k\in {\mathbb {Z}}^*}\) and \(\big ( ({\widetilde{\lambda }}_{k})^{-\gamma } {\widetilde{\Phi }}_k\big )_{k\in {\mathbb {Z}}^*}\) are equivalent basis in \({\widetilde{X}}_\gamma \), we have that
5.3 Characterization of the controllable spaces
Now we have all the ingredients needed to study the regularity properties of the controllable initial data with only one control. Let \({{\mathcal {H}}}_1\) be the subspace of \(X^*\) defined by
where \(\delta _k=|\lambda _k^1-\lambda _k^2|\) and \(\left( \lambda _k^s\right) _{k\in {\mathbb {Z}}^*,\, s\in \{1,2\}}\) are given in Proposition 3.1.
According to Theorem 4.3, the space \({{\mathcal {H}}}_1\) represents the set of controllable initial data of (1.1) with one control only, \(h^1\in L^2(0,T)\), acting on the extremity \(x=L\) of the first string. Our aim is to provide a characterization of this space in terms of Sobolev spaces. We remark that in the definition of the space \({{\mathcal {H}}}_1\) the coefficients \(a_k^1\) and \(a_k^1-a_k^2\) are multiplied by different weights. We shall see that, as a consequence of this property, the controllable initial data corresponding to the second string are more regular than the ones corresponding to the first string. Firstly, we have to introduce some preliminary results.
Let \(\{P_k\}_{k\in {\mathbb {Z}}^*}\cup \{Q_k\}_{k\in {\mathbb {Z}}^*}\) be two sequences of functions from X defined by
where \(\left( \Phi _k^j\right) _{k\in {\mathbb {Z}}^*,\,\, j\in \{1,2\}}\) are the eigenfunctions of the operator (D(A), A) introduced in (3.5) from Proposition 3.1. Also, we shall use the notation
where \({\widehat{\Phi }}_k^1\) and \({\widehat{\Phi }}_k^2\) have been defined by (5.14). We have the following result.
Theorem 5.1
\({{\mathcal {H}}}_1\) is a closed subspace of \(X^*\) and \(\{P_k\}_{k\in {\mathbb {Z}}^*}\cup \{Q_k\}_{k\in {\mathbb {Z}}^*}\) is a Riesz basis of \({{\mathcal {H}}}_1\). Moreover, (2.5) is well-posed in this space.
Proof
We recall that the dual space \(X^*\) is given by (5.9). Firstly, we remark that any \(U^0\in {{\mathcal {H}}}_1\) can be written as follows
On the other hand, for any \((a_k^1)_{k\in {\mathbb {Z}}^*}\) and \((a_k^2)_{k\in {\mathbb {Z}}^*}\) in \(\ell ^2\), we have that
The above relations imply that \(\{P_k\}_{k\in {\mathbb {Z}}^*}\cup \{Q_k\}_{k\in {\mathbb {Z}}^*}\) is a Riesz basis of \({{\mathcal {H}}}_1\).
Let now \(Z_0=\sum _{k\in {\mathbb {Z}}^*}\left( a_k^1 \Phi _k^1 + a_k^2 \Phi _k^2\right) \in {{\mathcal {H}}}_1\). Then, the corresponding solution of (2.5) is given by
and we have that
from which we deduce immediately that \(Z\in C([0,T],{\mathcal H}_1)\). \(\square \)
The main result of this section is the following one which provides a characterization of the space of controllable initial data \({{\mathcal {H}}}_1\). It asserts that the second component of an element from \({{\mathcal {H}}}_1\) is more regular than the first component.
Theorem 5.2
The following characterization of the space \({{\mathcal {H}}}_1\) holds
where the spaces \({\widetilde{X}}_{-1}\) and \({\widetilde{X}}_2\) are defined by (5.12) and (5.6), respectively. They are characterized in terms of Sobolev spaces by Propositions 5.2 and 5.1.
Proof
Let \(U^0=\left( \begin{array}{c}U_1^0\\ U_2^0\end{array}\right) \in {{\mathcal {H}}}_1\). According to Theorem 5.1, there exist \((\beta _k^1)_{k\in {\mathbb {Z}}^*}\) and \((\beta _k^2)_{k\in {\mathbb {Z}}^*}\) in \(\ell ^2({\mathbb {C}})\) such that
It follows that the second component \(U_2^0\) of \(U^0\) is given by
To show that \(U_2^0\in {\widetilde{X}}_2\), we evaluate each of its four components and we use the characterization of \({\widetilde{X}}_2\) in terms of Sobolev spaces given by Proposition 5.1. Firstly, let us show that the first component of \(U_2^0\), which according to the above relations is given by
belongs to \(H^3(0,L)\). To prove this property, we use that
\((\beta _k^1)_{k\in {\mathbb {Z}}^*}\in \ell ^2({\mathbb {C}})\) and Corollary 1 in “Appendix” in order to obtain that
On the other hand, from Ingham’s inequality [9] and the fact that \((\beta _k^1)_{k\in {\mathbb {Z}}^*},\, (\beta _k^2)_{k\in {\mathbb {Z}}^*}\in \ell ^2({\mathbb {C}})\), we deduce that
and
From relations (5.27)–(5.29), we obtain that \(U_{2,1}^0\in H^3(0,L)\). The rest of the components of \(U_2^0\) can be similarly analyzed. Also, notice that \(U_{2}^0\) fulfills the same boundary conditions as the eigenfunctions \({\widehat{\Phi }}_k^j\). Finally, we deduce that \(U_2^0\in {\widetilde{X}}_2\).
On the other hand, the first component \(U^0_1\) of \(U^0\) is given by
Since \((\beta _k^1)_{k\in {\mathbb {Z}}^*},\, (\beta _k^2)_{k\in {\mathbb {Z}}^*}\in \ell ^2\), according to Proposition 5.4, we have that \(U_1^0\in {\widetilde{X}}_{-1}\). We have proved that \({{\mathcal {H}}}_1\subset {\widetilde{X}}_{-1}\times {\widetilde{X}}_{2}\).
To show the inverse inclusion, let \(U^0=\left( \begin{array}{c}U_1^0\\ U_2^0\end{array}\right) \in {\widetilde{X}}_{-1}\times {\widetilde{X}}_2.\) Since \(U^0\in X^*\), from Proposition 3.2 it follows that there exist two sequences \((\beta _k^j)_{k\in {\mathbb {Z}}^*}\) such that \(\left( \frac{\beta _k^j}{\lambda _k^j}\right) _{k\in {\mathbb {Z}}^*}\in \ell ^2({\mathbb {C}})\), for \(j\in \{1,2\}\), such that \( U^0=\sum _{k\in {\mathbb {Z}}^*}\left( \beta _k^1 \Phi ^1_{k} + \beta _k^2 \Phi ^2_{k}\right) . \) We remark that
As in the first part of the proof, by taking into account the particular form of the function \(-{\widehat{\Phi }}_k^1+{\widehat{\Phi }}_k^2\) and the fact that \(\left( \frac{\beta _k^2}{\lambda _k^1}\right) _{k\in {\mathbb {Z}}^*}\in \ell ^2({\mathbb {C}})\), we deduce that
From the above relation, the fact that \(U^0_2\in {\widetilde{X}}_2\) and (5.30), we deduce that
By taking into account that, according to Proposition 5.4, the sequences \(\left( \delta _k \lambda _k^1{\widehat{\Phi }}_k^1\right) _{k\in {\mathbb {Z}}^*}\) is a Riesz basis in \({\widetilde{X}}_2\), the above relation is equivalent to
Relation (5.31), combined with the fact that \(\left( \frac{\beta _k^1}{\lambda _k^1}\right) _{k\in {\mathbb {Z}}^*}\in \ell ^2({\mathbb {C}})\), implies that \(U^0\in {{\mathcal {H}}}_1\), which completes the proof of the theorem. \(\square \)
Remark 5.5
Relation (5.26) shows that, in order to be controllable with a control \(h^1\in L^2(0,T)\) acting on the extremity \(x=L\) of the first string, the second component of the initial data have to be more regular than the first one. Hence, while the first component \(U_0^1\) is in \(L^2(0,L)\times {\mathbb {C}}\times {\widetilde{Y}}_{\frac{1}{2}}^*\), the second one, \(U_0^2\), belongs to \(H^3(0,L)\times H^2(0,L)\times {\mathbb {C}}^2\). This asymmetry is due to the presence of the masses at the extremities \(x=0\) of the strings which produce a regularization of the vibrations transmitted from the first string to the second one. The same phenomenon has been detected in other models joining masses and strings or beams [4, 5, 8]. However, this is the first result showing the asymmetry property when a spring is added to the system.
6 Comments on some limit cases
The aim of this section is to discuss the behavior of the system when its parameters (stiffness of the spring or mass of the bodies) tend to some limit values.
6.1 The case \(\kappa =0\)
We remark that \(\kappa =0\) corresponds to a system of two decoupled string equations (no spring, therefore, no connection). It is plausible that, under these circumstances, we cannot control the system with one control only. Firstly, we remark that formulas (3.3) holds even for \(\kappa =0\). Hence, in this case, the spectrum of the operator (D(A), A) given in (2.8) consists of a single family of eigenvalues \((i\mu _k^0)_{k\in {\mathbb {Z}}^*}\), satisfying the following asymptotic formula
In this case, each eigenvalue \(i\mu _k^0\) has geometric multiplicity two with two corresponding independent eigenvectors given by
where \(\alpha _k^0= \left| \mu ^0_k\right| \sqrt{ 2L +2 \sin ^2 (\mu ^0_k L)}\).
Taking into account that the strings are uncoupled, we deduce that we can control each of them separately by using two different controls but we cannot control one string through a control localized in the other one. As it can be deduced from Lemma 4.1, only particular initial data (those verifying \(a_k^1=a_k^2\), \(k\in {\mathbb {Z}}^*\) or, equivalently, when the initial data of the second string are identically zero) could be controllable in this case.
Remark 6.1
Here we give another view to understand the impact of vanishing stiffness of the spring (\(\kappa \rightarrow 0\)) on the null controllability problem by a direct constructive method. We consider the linear coupled system (1.1) with small \(\kappa \).
Now, we suppose that, in system (1.1), \(h^1(t)\) is a prescribed boundary function, while \(h^2(t)\) is the only control to be determined. In the framework of classical solutions, one way to solve this null control problem is to find a solution in the time-space domain such that the solution meets with given initial data, null final data and given boundary condition.
Take \(T>4L\) and \(\epsilon >0\) such that \(T_1:=\frac{T}{2}-\epsilon >2L\). Then, for any given initial data
with \((u^i_0(\cdot ), u^i_1(\cdot ))\in C^2[0,L]\times C^1[0,L], i=1,2\), the required solution can be constructed in the following steps:
-
In the domain \({\mathcal {R}}_f(T_1)=\{(t,x)|0\le t\le T_1, 0\le x\le L\}\), by solving the problem (1.1) (in which we take \(h^2(t)=0\)) and initial condition (6.3), we get a forward solution \(U_f:=(u^1_f, u^2_f)(t,x)\) satisfying the estimate
$$\begin{aligned} \Vert U_f\Vert _{C^2[{\mathcal {R}}_f(T_1)]} \le C_f \left( \sum _{i=1,2} \Vert \left( u^i_0, u^i_1\right) \Vert _{C^2[0,L]\times C^1[0,L]}+ \Vert h^1\Vert _{C^2[0,T_1]}\right) , \end{aligned}$$(6.4)where \(C_f\) is a positive constant.
-
In the domain \({\mathcal {R}}_b(T_1)=\{(t,x)|T-T_1\le t\le T, 0\le x\le L\}\), by solving the problem (1.1) (in which we take \(h^2(t)=0\)) and null final data \((u^i(T,x), u^i_t(T,x))= (0,0), i=1,2, 0\le x\le L\), we get a backward solution \(U_b:=(u^1_b, u^2_b)(t,x)\) satisfying the estimate
$$\begin{aligned} \Vert U_b\Vert _{C^2[{\mathcal {R}}_b(T_1)]} \le C_b \left( \Vert h^1\Vert _{C^2[T-T_1,T]}\right) , \end{aligned}$$(6.5)where \(C_b\) is a positive constant.
-
Then, we change the role of t and x in the system (1.1) and take
$$\begin{aligned} u^i(0,x)=u^i_0(x), \quad u^i(T,x)=0, \quad 0\le x\le L, i=1,2 \end{aligned}$$(6.6)as two new ’boundary’ conditions to the sidewise problem in the domain \({\mathcal {R}}(T)=\{(t,x)|0\le t\le T, 0\le x\le L\}\).
-
The sidewise solution \(u^1=u^1(t,x)\) is obtained by solving (1.1) and (6.6) (where we take \(i=1\)) with the new ’initial’ condition \((u^1(t,L), u^1_x(t,L))=(h^1(t), {\bar{h}}^1(t)), 0\le t\le T,\) where \(h^1(t)\) is the prescribed boundary function, while \({\bar{h}}^1(t)\in C^1[0,T]\) equals to the trace of \(u^1_{fx}\) and \(u^1_{bx}\) on time interval \([0,T_1]\) and \([T-T_1, T]\), respectively. Thus, \(u^1\equiv u^1_f\) on the intervals \(\{(t,x)| t=0, 0\le x\le L\}\) and \(\{(t,x)| 0\le t\le L, x=0\}\), while \(u^1\equiv u^1_b\) on the intervals \(\{(t,x)| t=T, 0\le x\le L\}\) and \(\{(t,x)| T-L\le t\le T, x=0\}\). We have
$$\begin{aligned} \Vert u^1(\cdot , 0), u^1_x(\cdot , 0)\Vert _{C^3[0,T]\times C^1[0,T]}\le & {} C_1 \left( \sum _{i=1,2} \Vert (u^i_0, u^i_1)\Vert _{C^2[0,L]\times C^1[0,L]}\right. \nonumber \\&\left. + \Vert h^1\Vert _{C^2[0,T]}\right) , \end{aligned}$$(6.7)where \(u^1(t,0)\) is a \(C^3\) function of time (called hidden regularity) and \(C_1\) is a positive constant.
-
The solution \(u^2=u^2(t,x)\) is obtained by solving the equation in (1.1) and (6.6) (\(i=2\)) with new ’initial’ data \((u^1(t,0), u^1_x(t,0))\) computed by the transmission conditions at the coupled side \(x=0\), that
$$\begin{aligned} \left\{ \begin{aligned} u^2(t,0)&=u^1(t,0)-\frac{1}{\kappa }(u^1_{tt}(t,0)-u^1_x(t,0)),\\ u^2_x(t,0)&=u^1_{tt}(t,0)-u^1_x(t,0)+u^2_{tt}(t,0),\\ \end{aligned} \quad 0\le t\le T. \right. \end{aligned}$$(6.8)Noting that \(u^1\) is solved in sidewise, which implies for \(L\le t\le 3L\), \(u^1(t,0) =\frac{1}{2}(h(t-L)+h(t+L))-\frac{1}{2}\int _{t-L}^{t+L}\bar{h}(\tau )\mathrm d\tau \), after derivation we obtain the above relations for \(u^1_x(t,0)\) and \(u^1_{tt}(t,0)\) which are determined by \((h^1(t), {\bar{h}}^1(t))\in C^2[0,T]\times C^1[0,T]\), where \(h^1(t)\) is independent of \(\kappa \). Thus, due to \(\frac{1}{\kappa }\) acting in the first equation above, for determined solution \(u^1\), as \(\kappa \rightarrow 0\), the norm \(|u^2(\cdot , 0)|_{C^0[0,T]}\rightarrow +\infty \) , then by the continuous dependence, the \(C^2\) norm of the corresponding sidewise solution \(u^2\), and the trace \(|u^2(t, L)|_{C^2[0,T]}\) will both go to infinity, which means that the boundary control \(h^2(t)\) will be extremely large when \(\kappa \) small enough.
It is worth mentioning here, due to the linearity of system (1.1), the solution will not blow up in the domain \({\mathcal {R}}(T)\) and we have one-sided controllability provided with enough control time (\(T>4L\)). However, the vanishing stiffness of spring will not influence the controllability time, but lead to large norm of desired control. On the other side, by classical arguments, it becomes clear that if we insist on a prescribed bound on the control, the controllability time will become infinite for \(\kappa \rightarrow 0\).
6.2 The case \(\kappa \rightarrow \infty \)
In this section, we want to see what does it happen when the rigidity of the spring becomes very high. We remark that the first family of the spectrum has the following asymptotic property, with respect to the parameter \(\kappa \).
Proposition 6.1
For each \(k\in {{\mathbb {N}}^*}\), the following asymptotic formula holds
Proof
In order to study the asymptotic behavior of the family \((\mu _k^1)_{k\ge 1}\), with respect to the parameter \(\kappa \), we are looking for a real constant \(\alpha \) such that
Using the asymptotic relation (6.10) in (3.7), we obtain that \(\alpha (\kappa )=-\frac{k\pi }{2L^2}\) and (6.9) holds true. \(\square \)
The asymptotic formula (6.10) shows that in the limit case \(\kappa =\infty \) the spectrum of the operator (D(A), A) consists of a double family of eigenvalues \((i\mu _k^{\infty ,1})_{k\in {\mathbb {Z}}^*}\cup (i\mu _k^{\infty ,2})_{k\in {\mathbb {Z}}^*}\), where \(\mu _k^{\infty ,2}=\mu _k^2\) and
Notice that the eigenvalues equation corresponding to the case a) in the proof of Proposition 3.1 is equivalent to
which, in the limit case \(\kappa =\infty \), is reduced to
In fact, when \(\kappa \) tends to infinity, the original system (2.5) changes into
which corresponds to a system consisting on two flexible strings connected at \(x=0\) by a kind of rigid joint (see Fig. 2).
System of two coupled strings with bodies and rigid joint
A similar model has been already studied in [4, 8] where, like in our case, asymmetric spaces of controllable initial data have been identified when a unique control acts on the extremity \(x=L\) of one of the two strings.
6.3 The case of vanishing mass
In (2.5), we have considered that the mass of the bodies attached at the extremities \(x=0\) are both equal to one. We could consider that the mass of these bodies is an arbitrary positive numbers M. In this case, the boundary conditions at \(x=0\) should be changed to
The eigenvalues equations become
Once again, the spectrum of the corresponding differential operator consists of two families of eigenvalues \((i\mu _k^{M,1})_{k\in {\mathbb {Z}}^*} \cup (i\mu _k^{M,2})_{k\in {\mathbb {Z}}^*}\), and the following asymptotic formulas hold, with respect to k,
However, (6.14) does not indicate the behavior of these eigenvalues as M goes to zero. In the limit case \(M=0\) the spectrum consists of two families of eigenvalues \((i\mu _k^{0,1})_{k\in {\mathbb {Z}}^*} \cup (i\mu _k^{0,2})_{k\in {\mathbb {Z}}^*}\), representing the roots of the following two equations:
We remark that, in this limit case, the boundary conditions (6.12) become
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Appendix
Appendix
In “Appendix,” we study the Riesz basis properties of some sequences of trigonometric functions in several spaces. The following lemma analyzes the \(\omega \)-independent property of these sequences.
Lemma 7.1
Let \((\mu _k)_{k\ge 1}\) a sequence of different positive numbers with the property that
Then, the sequences \(\left( \sin \left( \mu _k x\right) \right) _{k\ge 2}\) and \(\left( \cos \left( \mu _k x\right) \right) _{k\ge 2}\cup \{1\}\) are \(\omega \)-independent in \(L^2(0,L)\). Moreover, the sequence \(\left( \frac{\sin \left( \mu _k (x-L)\right) }{\mu _k}\right) _{k\ge 1}\) is \(\omega \)-independent in \(H^1_L(0,L)\), the sequence \(\left( \frac{\sin \left( \mu _k (x-L)\right) }{\mu _k^2}\right) _{k\ge 1}\) is \(\omega \)-independent in \(H^2(0,L)\cap H^1_L(0,L)\) and the sequence \(\left( \frac{\sin \left( \mu _k (x-L)\right) }{\mu _k^3}\right) _{k\ge 1}\) is \(\omega \)-independent in the space \(\left\{ u\in H^3(0,L)\cap H^1_L(0,L): u_{xx}(L)=0\right\} \).
Proof
By taking into account (7.1), it follows that, given \(\lambda \in (0,1)\), the following inequality holds
for any \(N\in {\mathbb {N}}^*\) sufficiently large, \(p\ge 1\) and scalars \((c_k)_{N\le |k|\le N+p}\subset {\mathbb {C}}\). Suppose that \(\left( \sin \left( \mu _{k} x\right) \right) _{k\ge 2}\) is not \(\omega \)-independent in \(L^2(0,L)\). Then, there exists a sequence of scalars \((a_k)_{k\ge 1}\), which are not all zero, such that
which is equivalent to
Let \(a_{k_0}\ne 0\). According to (7.2), we can suppose that \(2\le k_0< N\). It follows that,
Hence, there exists an exponential \(e^{-i\mu _{k_0} x}\) which is a combination of the others. According to [20, Theorem 8, Ch. 3], it follows that \(\left( e^{-i\mu _k x}\right) _{k\ge 2,\, k\ne k_0}\cup \left( e^{i\mu _k x}\right) _{k\ge 2}\) is complete in \(L^2(-L,L)\). Since the completeness is unaffected when a finite number of exponentials are replaced by different terms, the family \(\Lambda _{k_0}:=\left( e^{-i\mu _k x}\right) _{k\ge N+1}\cup \left( e^{i\mu _k x}\right) _{k\ge N+1}\cup \left( e^{i\frac{k\pi }{L} x}\right) _{0< |k|< N,\, k\ne -k_0}\) is complete in \(L^2(-L,L)\). On the other hand, from (7.2) and the Paley–Wiener stability criterion (see [20, Theorem 10, Ch. 1]), we deduce that \(\left( e^{-i\mu _k x}\right) _{k\ge N+1}\cup \left( e^{i\mu _k x}\right) _{k\ge N+1}\cup \left( e^{i\frac{k\pi }{L} x}\right) _{0\le |k|< N}\) is a Riesz basis in \(L^2(-L,L)\) which contradicts the completeness of \(\Lambda _{k_0}\). Hence, the sequence \(\left( \sin \left( \mu _k x\right) \right) _{k\ge 2}\) is \(\omega \)-independent in \(L^2(0,L)\). The case of the sequence \(\left( \cos \left( \mu _k x\right) \right) _{k\ge 2}\cup \{1\}\) can be similarly studied. Now, let us show that the sequence \(\left( \frac{1}{\mu _k}\sin \left( \mu _k (x{-}L)\right) \right) _{k\ge 1}\) is \(\omega \)-independent in \(H^1_L(0,L)\). Let us suppose that \(\left( \frac{1}{\mu _k}\sin \left( \mu _k (x-L)\right) \right) _{k\ge 1}\) is not \(\omega \)-independent in \(H^1_L(0,L)\). There exists a sequence of scalars \((a_k)_{k\ge 1}\), which are not all zero, such that
which implies that
We already know that \(\left( \cos \left( \mu _k (x-L)\right) \right) _{k\ge 2}\cup \{1\}\) is \(\omega \)-independent in \(L^2(0,L)\). Consequently, the only possibility is to have \(a_1\ne 0\). It follows that
Consequently, \(e^{i \mu _1 (x-L)}\in \overline{\hbox {Span}\{\left( e^{i \mu _k (x-L)}\right) _{k\ge 2}\cup \left( e^{-i \mu _k (x-L)}\right) _{k\ge 1}\}}\) in \(L^2(-L,L)\), which implies that \(\left( e^{i \mu _k (x-L)}\right) _{k\ge 2}\cup \left( e^{-i \mu _k (x-L)}\right) _{k\ge 1}\) is complete in \(L^2(-L,L)\). An argument similar to the one used in the first part of the proof leads us to a contradiction. Hence, the sequence \(\left( \frac{1}{\mu _k}\sin \left( \mu _k (x-L)\right) \right) _{k\ge 1}\) is \(\omega \)-independent in \(H^1_L(0,L)\). To show that \(\left( \frac{\sin \left( \mu _k (x-L)\right) }{\mu _k^2}\right) _{k\ge 1}\) is \(\omega \)-independent in \(H^2(0,L)\cap H^1_L(0,L)\) and \(\left( \frac{\sin \left( \mu _k (x-L)\right) }{\mu _k^3}\right) _{k\ge 1}\) is \(\omega \)-independent in the space \(\left\{ u\in H^3(0,L)\cap H^1_L(0,L): u_{xx}(L)=0\right\} \) we proceed as above, by taking two or three derivatives instead of one. \(\square \)
We pass to study the Riesz basis properties of the trigonometric sequences mentioned in the previous lemma.
Lemma 7.2
Let \((\mu _k)_{k\ge 1}\) be a sequence of different positive numbers verifying (7.1) and define the spaces
where \(d(s)=2\) if \(s=0\) and \(d(s)=1\) if \(s\ge 1\). The following properties hold:
For each \(s\in \left\{ 0,1,2,3\right\} \), \({{\mathcal {X}}}_s\) is a Hilbert space and the set \(\left( \frac{1}{(\mu _k)^s}\sin \left( \mu _k(x-L)\right) \right) _{k\ge d(s)}\) is a Riesz basis in \({{\mathcal {X}}}_s\).
Proof
Let us deal in detail with the case \(s=0\). The rest of the cases can be studied in a similar way. We show that \(\left( \sin \left( \mu _k(x-L)\right) \right) _{k\ge 2}\) is a Riesz basis in \(L^2(0,L)\). Firstly, we remark that \(\sin \left( \mu _k(x-L)\right) \) is a function from \(L^2(0,L)\). Moreover, we have that:
-
\(\left( \sin \left( \frac{k\pi }{L}(x-L)\right) \right) _{k\ge 1}\) is an orthonormal basis in \(L^2(0,L)\).
-
Since \(\left| \mu _{k+1}-\frac{k\pi }{L}\right| \sim \frac{1}{k\pi }\) as \(k\rightarrow \infty \), it follows that
$$\begin{aligned} \sum _{n\ge 1} \left\| \sin \left( \mu _{k+1}(x-L)\right) -\sin \left( \frac{k\pi }{L}(x-L)\right) \right\| _{L^2(0,L)}^2<\infty . \end{aligned}$$(7.5) -
According to Lemma 7.1, the sequence \(\left( \sin \left( \mu _k x\right) \right) _{k\ge 2}\) is \(\omega \)-independent in \(L^2(0,L)\).
From these properties and Bari’s Theorem [20, Theorem 15, Ch. 1], we deduce that \(\left( \sin \left( \mu _k(x-L)\right) \right) _{k\ge 2}\) is a Riesz Basis in \(L^2(0,L)\). Consequently, \({{\mathcal {X}}}_0=L^2(0,L)\). \(\square \)
Remark 7.1
Lemma 7.2 holds for the sequences \(\left( {\tilde{\mu }}_k\right) _{k\ge 1}\), \(\left( \mu _k^j\right) _{k\ge 1}\), \(j\in \{1,2\}\) introduced in Sects. 3 and 5. Indeed, the sequences \(\left( \mu _k^1\right) _{k\ge 1}\), \(\left( \mu _k^2\right) _{k\ge 1}\) and \(\left( {\tilde{\mu }}_k\right) _{k\ge 1}\) verify the asymptotic behavior (7.1).
We conclude “Appendix” with the following two results.
Lemma 7.3
Let \((\mu _k^1)_{k\in {\mathbb {Z}}^*}\) and \((\mu _k^2)_{k\in {\mathbb {Z}}^*}\) be the real families given in Proposition 3.1. Then, there exists a positive constant C such that, for any \((\beta _k)_{k\in {\mathbb {Z}}^*}\) in \(l^2\), the following inequality holds
Proof
Using Taylor decomposition theorem, we have
thus, using in addition Cauchy–Schwarz inequality and Ingham’s inequality, we infer the existence of a constant \(C>0\) such that
where for the last inequality we have used \(|\mu _k^1-\mu _k^2|\sim \frac{1}{k^3}\) as \(k\rightarrow \infty \). \(\square \)
Corollary 1
In the same hypothesis as in Lemma 7.3, we have
Proof
From (7.6), where we have chosen \((\Re \beta _k)_{k\in {\mathbb {Z}}^*}\) instead of \((\beta _k)_{k\in {\mathbb {Z}}^*}\), we get
Using in addition similar estimates for \((\Im \beta _k)_{k\in {\mathbb {Z}}^*}\), we infer inequalities (7.7) and (7.8).
\(\square \)
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Leugering, G., Micu, S., Rovenţa, I. et al. Controllability properties of a hyperbolic system with dynamic boundary conditions. J. Evol. Equ. 22, 65 (2022). https://doi.org/10.1007/s00028-022-00823-5
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DOI: https://doi.org/10.1007/s00028-022-00823-5
Keywords
- Wave equation with dynamic boundary conditions
- Exact boundary controllability
- Non-harmonic Fourier analysis