Abstract
In this paper, we consider the energy decay of a damped hyperbolic system of two degenerate wave equations coupled by velocities when only one equation is directly damped by a linear boundary feedback. To this aim, we first prove that the proposed system is well-posed using the semigroup theory. Then, under the hypothesis that the coupling coefficient is positive and small, we show that the total energy of the whole system decays exponentially. The explicit energy decay rate is established by using the energy multiplier method.
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1 Introduction
The stabilization problems for scalar wave equations have received considerable attention in the literature, with numerous contributions achieved over the past several years. We refer, for example, to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] and the articles citing them.
Recently, the subject of indirect stabilization of coupled wave equations has received a lot of attention of many authors. This notion introduced by Russell [17] concerns stabilization questions for coupled equations with a reduced number of feedbacks. This means that some equations of the coupled system are not directly damped, but one then hopes that the coupling effects will be sufficient so that the full system is stabilized. This kind of damped systems is very important from the applications in control theory point of view, since it may be impossible or too expensive to damp each equation because of engineering or biological constraints.
For this reason, the theory of indirect stabilization for coupled wave equations has been investigated extensively. See, in particular [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] and the rich references therein. We also refer to the two monographs [35] and [36] for a comprehensive review.
However, all the previous results concern nondegenerate problems. On the other hand, from [37, Theorem 4.5], we know that the linearly damped scalar degenerate wave equation is exponentially stable.
In this paper, the question we are interested in is to determine if it is still possible to achieve the exponential stability of a system of two degenerate waves by means of only one damping.
More precisely, for given \(\beta > 0\), we investigate the stabilization of the following model of coupled degenerate wave equations with only one boundary damping:
where \(a\in C([0,1]) \cap C^1((0,1])\) is positive on ]0, 1] but vanishes at zero, \(b>0\) is the coupling parameter and
In view of the results in [38], since the coupling acts here in a stronger way (through velocities), we are interested in finding conditions on the parameters of the system such that the energy of this linearly damped system decays exponentially. Indeed, different from the case of couplings through displacements [18,19,20], it is shown in [38] that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities.
Here, in agreement with [38], our main result (see theorem 9) asserts that a single feedback is sufficient to guarantee that the energy of the full system (1) decays exponentially to 0 at infinity. This extends the energy decay result in [37] for the single degenerate wave equation to the system of two degenerate wave equations which are coupled through the velocities.
To the best of our knowledge, this is the first paper where the asymptotic behavior of solutions to strongly coupled degenerate wave equations is studied with the main particularity that only one equation is effectively damped by a boundary feedback acting on one end only.
In order to study the damped system (1), we will assume that the coupling parameter b is sufficiently small and the function a satisfies the following assumptions:
where \([\cdot ]\) stands for the integer part.
The rest of the paper is organized as follows. In Sect. 2, we introduce the appropriate functional spaces that are naturally associated with degenerate problems and preliminary results used throughout the paper. Section 3 is devoted to the proof of the well posedness of the considered system. In Sect. 4, we study the boundary stabilization problem proving its exponential stability.
2 Preliminary results
Let \(a \in \mathcal {C}([0,1]) \cap \mathcal {C}^{1}(]0,1])\) be a function satisfying assumptions (2). At first, as in [37], we introduce some weighted Sobolev spaces that are naturally associated with degenerate operators. We denote by \(H_{a}^{1}(0,1)\) the space of all functions \(u \in L^{2}(0,1)\) such that
It is easy to see that \(H_{a}^{1}(0,1)\) is a Hilbert space with the scalar product
and associated norm
Next, we define
Note that if \(u \in H_{a}^{2}(0,1)\), then \(au^\prime \) is continuous on [0, 1].
In the following proposition, we collect useful properties of the above functional spaces which will play an important role in order to evaluate several boundary terms, see [37, Proposition 2.5].
Proposition 1
Assume that a is a function satisfying (2). Then the following assertions hold true:
-
1.
For every \(u \in H_{a}^{1}(0,1)\)
$$\begin{aligned} \lim _{x \downarrow 0} x |u(x)|^2=0. \end{aligned}$$(3)Moreover, if \(\mu _a \in [0,1[\), then u is absolutely continuous in [0, 1].
-
2.
For every \(u \in H_{a}^{2}(0,1)\)
$$\begin{aligned} \lim _{x \downarrow 0} x a(x) |u^{\prime }(x)|^2=0 . \end{aligned}$$(4) -
3.
For all \(u \in H_{a}^{2}(0,1)\) and for all \(v \in H_{a}^{1}(0,1)\)
$$\begin{aligned} \lim _{x \downarrow 0} a(x) u^{\prime }(x) v(x)=0, \end{aligned}$$(5)assuming, in addition, \(v(0)=0\) if \(\mu _{a} \in [0,1[\).
-
4.
If \(\mu _a\in [1,2[\), for every \(u\in H_{a}^{2}(0,1)\)
$$\begin{aligned} \lim _{x \downarrow 0} a(x) u^{\prime }(x) =0. \end{aligned}$$(6)
In view of Proposition 1, we see that the boundary conditions imposed at \(x=0\) make sense for any classical solution of (1). Such conditions are of Dirichlet type if \(\mu _{a}\in [0,1[\), whereas they are of Neumann/Dirichlet type at \(x=0\) and \(x=1\), respectively, if \(\mu _{a}\in [1,2[\).
In order to express the boundary conditions of the first component of the solution of (1) in the functional setting, we define the space \(H_{a,0}^{1}(0,1)\) depending on the value of \(\mu _{a}\), as follows:
- (i):
-
For \(0 \le \mu _{a}<1\), we define
$$\begin{aligned} H_{a,0}^{1}(0,1):=\left\{ u \in H_{a}^{1}(0,1) \mid u(0)=u(1)=0\right\} . \end{aligned}$$ - (ii):
-
For \(1 \le \mu _{a}<2\), we define
$$\begin{aligned} H_{a,0}^{1}(0,1):=\left\{ u \in H_{a}^{1}(0,1) \mid u(1)=0\right\} . \end{aligned}$$
Let us recall the following version of Poincaré’s inequality, which is proved in [37, Proposition 2.2].
Lemma 2
Assume (2) holds. Then
where
Then set
which, thanks to Lemma 2, defines a norm on \(H_{a,0}^{1}(0,1)\) that is equivalent to \(\Vert \cdot \Vert _{H^{1}_{a}(0,1)}\).
Finally, we define
Observe that all functions \(u \in H_{a,0}^{2}(0,1)\) satisfy the above homogeneous boundary conditions at both \(x=0\) and \(x=1\).
3 Well-posedness
In this section, we first provide existence and uniqueness results of solutions for the damped hyperbolic system (1). Let us denote by \(W_{a}^{1}(0,1)\) the space \(H_{a}^{1}(0,1)\) itself if \(\mu _a \in \left[ 1,2\right) \) and, if \(\mu _a \in \left[ 0,1)\right. \), the closed subspace of \(H_{a}^{1}(0,1)\) consisting of all the functions \(u \in H_{a}^{1}(0,1)\) such that \(u(0)=0.\) Moreover, we set
Note that \(W_{a}^{2}(0,1)=H_{a}^{2}(0,1)\) when \(\mu _{a} \in [1,2).\)
In the Hilbert space \(W_{a}^{1}(0,1)\), we consider the following inner product
and the associated norm
First of all, recall the following preliminary results (see [37, Proposition 2.5 and Proposition 4.3]).
Proposition 3
Assume (2) holds. Then
We also have
where
In view of (10), observe that
defines a norm on \(W_a^1(0,1)\) that is equivalent to \(\Vert \cdot \Vert _{W_{a}^{1}(0,1)}\).
In order to prove the well-posedness of the system (1) and establish an exponential decay result, we will need the following (see [37, Proposition 4.3]).
Proposition 4
Assume (2) holds and \(\beta > 0\). Then, we have
where
Moreover, we also have
where
Now, let us define the energy space \(\mathcal {H}_{a}^{\beta }\) by
It is easy to see that \(\mathcal {H}_{a}^{\beta }\) is a Hilbert space, equipped with the scalar product defined by
for all \(U=(u_{1}, u_{2}, u_{3}, u_{4}), \widetilde{U}=(\widetilde{u}_{1}, \widetilde{u}_{2}, \widetilde{u}_{3}, \widetilde{u}_{4}) \in \mathcal {H}_{a}^{\beta }\). The expression \(\Vert \cdot \Vert _{\mathcal {H}_{a}^{\beta }}\) will denote the corresponding norm.
We are now ready to study the well posedness of system (1) by using semigroup theory. For this, we define the unbounded linear operator \(\mathcal {A}_{a}^{\beta }: D(\mathcal {A}_{a}^{\beta })\subset \mathcal {H}_{a}^{\beta } \rightarrow \mathcal {H}_{a}^{\beta }\) by
and
Setting \(U(t)=(v(t),v_t(t),u(t),u_t(t))\), then system (1) can be transformed into the first order evolution equation on the Hilbert space \(\mathcal {H}_{a}^{\beta }\) as follows
where \(U(0)=(v_0,v_1,u_0,u_1)\).
In view of Proposition 1, if \(U=(v, v_t, u, u_t) \in D\left( \mathcal {A}_{a}^{\beta }\right) \), then U satisfies the boundary conditions \(Bv(0)=Bu(0)=0\) at \(x=0\) and the Dirichlet boundary condition \(v(1)=0\) at \(x=1\). Notice also that \(u_{x}(1)\), \(u_t(1)\) and \(\beta u(1)\) are well defined for all \(U=(v, v_t, u, u_t) \in D\left( \mathcal {A}_{a}^{\beta }\right) \) because of the classical Sobolev embedding theorem.
The next result holds.
Proposition 5
Assume (2) holds and consider \(\beta >0\). Then \(\mathcal {A}_{a}^{\beta }\) is a maximal dissipative operator on \(\mathcal {H}_{a}^{\beta }\).
Proof
For all \(U=(u_{1},u_{2},u_{3},u_{4})\in D(\mathcal {A}_{a}^{\beta })\), we have
Integrating by parts and using (5), we get
which implies that \(\mathcal {A}_{a}^{\beta }\) is dissipative. In order to show that \(\mathcal {A}_{a}^{\beta }\) is maximal dissipative, it remains to prove that \(R(I-\mathcal {A}_{a}^{\beta })=\mathcal {H}_{a}^{\beta }\). Let \(F=\left( f_{1}, f_{2}, f_{3}, f_{4}\right) \in \mathcal {H}_{a}^{\beta }\). We look for an element \(U=\left( u_{1}, u_{2}, u_{3}, u_{4}\right) \in D(\mathcal {A}_{a}^{\beta })\) such that
Suppose that we have found \(u_1\) and \(u_3\) with the appropriate regularity. Therefore, the first and the third equations in (16) give
Then, it is clear that \(u_2\in H^{1}_{a,0}(0,1)\) and \(u_4\in W^{1}_{a}(0,1)\). By using (16) and (17) the functions \(u_1\) and \(u_3\) satisfy the following system:
Solving system (18) is equivalent to finding \((u_1,u_3)\in H^{2}_{a,0}(0,1)\times W^{2}_{a}(0,1)\) such that
for all \((\phi _{1},\phi _{2})\in C^\infty _{c}(0,1)\times C^\infty _{c}(0,1)\).
To this aim, introduce the bilinear form \(\Lambda :\left( H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\right) ^{2} \rightarrow \mathbb {R}\) given by
and the linear form \(L: H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\rightarrow \mathbb {R}\) given by
From (7), (13) and the definition of \(|\cdot |_{W_a^1(0,1)}\), one can show that \(\Lambda \) is a continuous bilinear form on \(H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\) and L is a continuous linear functional on \(H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\). Furthermore, it is easy to see that \(\Lambda \) is also coercive on \(H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\). As a consequence, by the Lax-Milgram Theorem, there exists a unique \(\left( u_{1}, u_{3}\right) \in H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\) such that
Now, we will prove that \(\left( u_{1}, u_{2}, u_{3}, u_{4}\right) \in D(\mathcal {A}_{a}^{\beta })\) and solves (16). Since \(C^\infty _{c}(0,1)\times C^\infty _{c}(0,1)\subset H_{a,0}^{1}(0,1)\times W_{a}^{1}(0,1)\), (20) holds for every \(\left( \phi _{1}, \phi _{2}\right) \in C^\infty _{c}(0,1)\times C^\infty _{c}(0,1)\). Hence, we have (19) which is equivalent to (18). This yields \((u_1,u_3)\in H_{a}^{2}(0,1)\times H_{a}^{2}(0,1)\) and thus \((u_1,u_3)\in H_{a,0}^{2}(0,1)\times W_{a}^{2}(0,1)\).
Coming back to (18), we deduce after an integration by parts, together with (5), that
This combined with (20) leads to:
Using the fact that \(a(1)>0\) and the function \(\phi _{2}\) defined by \(\phi _{2}(x) = x\) for all \(x\in (0,1)\) is in \(W^{1}_{a}(0, 1)\), we infer that
Finally, recalling (17), we deduce that \(\left( u_{1}, u_{2}, u_{3}, u_{4}\right) \in D(\mathcal {A}_{a}^{\beta })\) and solves (16). The proof is thus complete. \(\square \)
By using the Hille-Yosida theorem (see [39, Theorem 4.5.1] or [40, Theorem A.7]), we deduce that the operator \(\mathcal {A}_{a}^{\beta }\) generates a \(C_0\)-semigroup of contractions \(\left( e^{t\mathcal {A}_{a}^{\beta }}\right) _{t\ge 0}\). The solution of the Cauchy problem (15) admits the following representation
which leads to the well-posedness of (15). Hence, we have the following result.
Corollary 6
Assume (2) holds and consider \(\beta >0\). For any \(U_{0} \in \mathcal {H}_{a}^{\beta }\), there exists a unique solution \(U \in C^{0}([0,+\infty ); \mathcal {H}_{a}^{\beta })\) of problem (15). Moreover, if \(U_{0}\in D(\mathcal {A}_{a}^{\beta })\), then
4 Exponential stability
In this section, we study the exponential stability of system (1). To this aim we first define its energy as
In particular, it is possible to prove that the energy is a non increasing function.
Lemma 7
Assume (2) holds. Let \(U=(v, v_t, u, u_t)\) be a regular solution of system (1). Then, the energy \(\mathcal {E}_{v,u}\) associated to (v, u) satisfies
Proof
Multiplying the first and the second equation of (1) by \(v_t\) and \(u_t\) respectively, integrating by parts over (0, 1), we get
and
Adding (23) and (24), by using the boundary conditions, we obtain
Using the fact that \(u_x(t,1)=-u_t(t,1)-\beta u(t,1)\), we get the desired equation (22). \(\square \)
From (22), it follows that system (1) is dissipative. Now we address the question how fast this energy decays. Precisely, we give an exponential stabilization estimate based on a direct application of the multiplier method.
Prior to the precise statement of our main result, we first recall the following result (See [37, Proposition 4.4]).
Proposition 8
Assume (2) holds and consider \(\beta >0\). Then, for every \(\lambda \in \mathbb {R}\), the variational problem
admits a unique solution \(z \in W_{a}^{1}(0,1)\) which satisfies the elliptic estimates
Moreover, \(z \in W_{a}^{2}(0,1)\) and solves
Let us also introduce the following notations:
and
Observe that, by condition (2) (ii) and assuming that \(0<b<b_a\), we have that \(M_{a,b}>0\).
Now, we are in position to state our main stability result.
Theorem 9
Assume (2) holds and that \(\beta >0\) is given. Suppose \(0<b<b_{a}\). Then for any \(U_{0}=\left( v_{0}, v_{1}, u_{0}, u_{1}\right) \in \mathcal {H}_{a}^{\beta }\), the solution \(U=(v,v_t,u,u_t)\) of (1) satisfies the uniform exponential decay
where \(M_{a, b,\beta }>0\) is given in (72) and is independent of \(U_{0}\).
Remark 1
As far as we know, this result on exponential stabilization seems to be new even for strongly coupled hyperbolic systems with nondegenerate variable coefficients.
The next lemmas are technical results to be used in the proof of Theorem 9 given below.
Lemma 10
Assume (2) holds and that \(\beta >0\) is given. Let \(U_{0}=\left( v_{0}, v_{1}, u_{0}, u_{1}\right) \in D(\mathcal {A}_{a}^{\beta })\) and \(U=(v,v_t,u,u_t)\) be the solution of (1). Then for every \(0\le S\le T\) and \(\varepsilon >0\) the following inequality holds:
Proof
Multiplying the second equation of (1) by \(v_t\), integrating by parts over \((S, T) \times (0,1)\), we get
because \(a(x)u_{x}(t,x) v_t(t,x)\) vanishes at \(x=1\) and, owing to (5), also at \(x=0\).
After integrating by parts on time, this gives
Next, we multiply the first equation of (1) by \(u_t\) and integrate the resulting equation over \((S,T )\times (0,1)\). This gives, after a suitable integration by parts,
because \(a(x)v_{x}(t,x) u_t(t,x)\) vanishes at \(x=0\) owing to (5).
Combining (31) and (32), we obtain
By the Young inequality, we have
From the fact that the energy is non-increasing, it follows that
On the other hand, for \(\varepsilon >0\), using Young’s inequality, we have
By the dissipation relation (22), we obtain
Thus, inserting (34) and (35) into (33), we get the required inequality (30). \(\square \)
Lemma 11
Assume (2) holds and that \(\beta >0\) is given. Let \(U_{0}=\left( v_{0}, v_{1}, u_{0}, u_{1}\right) \in D(\mathcal {A}_{a}^{\beta })\) and \(U=(v,v_t,u,u_t)\) be the solution of (1). Then for all \(0\le S\le T\) and all \(\varepsilon >0\) the following inequality holds:
where
and
Proof
We multiply the second equation of (1) by \(2x u_{x}\) and we integrate by parts over \((S,T)\times (0,1)\) as follows:
On the other hand, by integrating by parts and owing to (3)–(4), we have
Inserting (40) into (39), we get
Now, we proceed by multiplying the second equation of (1) by u and integrating by parts over \((S, T) \times (0,1)\), to get
Using the boundary conditions together with (5), this gives
By adding to (41) the identity (42) multiplied by \(\frac{\mu _{a}}{2}\), we obtain
where the function \(h_0\) is given by
Observe that, by the definition of \(\mu _a\), we have
This, combined with (43), gives
On the other hand, using Young’s inequality, we have
Next, we compute:
By using the boundary conditions, (3) and (57), we obtain that
Combining (45) and (46), we get
where \(d^{1}_{a,\beta }\) is given in (37).
Moreover, we have
Finally, inserting (47) and (48) into (44), one obtains the desired estimate (36). \(\square \)
Lemma 12
Assume (2) holds and that \(\beta >0\) is given. Let \(U_{0}=\left( v_{0}, v_{1}, u_{0}, u_{1}\right) \in D(\mathcal {A}_{a}^{\beta })\) and \(U=(v,v_t,u,u_t)\) be the solution of (1). Then for all \(0\le S\le T\) the following inequality holds:
where \(C_{a}=\frac{1}{a(1)} \min \left\{ 4, \frac{1}{2-\mu _{a}}\right\} .\)
Proof
We multiply the first equation of (1) by v and integrate the resulting equation over \((S,T)\times (0, 1)\). After suitable integrations by parts, this gives
Using the fact that \(a(x)v_x(t,x)v(t,x)\) vanishes at \(x=1\) and, owing to (5) also at \(x=0\), we get
On the other hand, using Young’s inequality and the Poincaré inequality (7), we have
Thus
One can show similarly that
Then, inserting (51) and (52) into (50), we arrive at the desired inequality (49). \(\square \)
Lemma 13
Assume (2) holds and that \(\beta >0\) is given. Let \(U_{0}=\left( v_{0}, v_{1}, u_{0}, u_{1}\right) \in D(\mathcal {A}_{a}^{\beta })\) and \(U=(v,v_t,u,u_t)\) be the solution of (1). Then for all \(0\le S\le T\) the following inequality holds:
Proof
We multiply the second equation of (1) by \(2x v_{x}\) and we integrate by parts over \((S,T)\times (0,1)\) as follows:
On the other hand, by integrating by parts and owing to (3)–(4), we have
Inserting (55) into (54), we get
Now, observe that condition (2) (ii) yields:
From Young inequality and (57), it follows that:
Finally, combining (56), (58), (59) and the inequality \(x|a^\prime (x)|<2a(x)\), one obtains (53). \(\square \)
Lemma 14
Assume (2) holds and that \(\beta >0\) is given. Let \(U_{0}=\left( v_{0}, v_{1}, u_{0}, u_{1}\right) \in D(\mathcal {A}_{a}^{\beta })\) and \(U=(v,v_t,u,u_t)\) be the solution of (1). Then for all \(0\le S\le T\) and all \(\delta >0\) the following inequality holds:
Proof
Set \(\lambda =u(t, 1)\) and let z be the solution of the degenerate elliptic problem (27). We multiply the second equation of (1) by z and integrate by parts the resulting equation over \((S, T) \times (0,1),\) to obtain
By using the boundary conditions at \(x=0\) together with (5), this gives
Then, from the boundary conditions at \(x=1\) in both systems (1) and (27), we get
Hence
It only remains to estimate in a suitable way the terms on the right-hand side of the previous inequality as follows. First, using Young’s inequality and thanks to the second inequality in (26), we have
Similarly, by Young’s inequality and the Poincaré inequality (7), we have
On the other hand, keeping in mind the second inequality in (26), we have
Furthermore, thanks to the first inequality in (26) and the definition of \(|\cdot |_{W_{a}^{1}(0,1)}\), we have
so that
Hence, by using the Young’s inequality
and taking into account the estimates (62)–(65) in (61), we deduce that
Finally, using (22) in the above estimate, one obtains (60). \(\square \)
Proof of Theorem 9
As usual, let us assume that \(U=(v,v_t,u,u_t)\) is a regular solution of (1) (the general case can be recovered by an approximation argument). We start by combining (49) multiplied by \(\frac{2-\mu _{a}}{2}\) and (36) multiplied by 2, to obtain that, for all \(0\le S\le T\) and all \(\varepsilon >0\),
which can be rewritten as
Moreover, from (30), we can see that
Inserting this last inequality into (67) and using the definition of \(M_{a,b}\) (see (28)), then we have
We now choose \(\varepsilon =\varepsilon _{a,b}:=\frac{(2-\mu _{a})M_{a,b}}{4\left( \frac{1}{2}+\frac{3(2-\mu _{a})}{8b}\right) (6+\frac{2b}{\min \{1,a(1)\}})}\) and use (53), then (68) becomes
We now estimate the last term on the right-hand side of this inequality where we recall that the function h is given in (38). We have that
where
Therefore, by using (60) with \(\delta =\delta _{a,b,\beta }:=\frac{\left( 2-\mu _{a}\right) M_{a,b}}{ 8\eta _2\left( 1+bC_{a}+\frac{1}{\beta ^{3}}\right) }\) and keeping in mind the dissipation relation (22), we get
Using (70) in (69), it results that
where
Finally, by [9, Theorem 8.1] (that has also been used before in [41]), we conclude that the energy of the system (1) satisfies the exponential decay estimate (29). The proof is thus complete. \(\square \)
Remark 2
We observe that the constant \(M_{a, b, \beta }\), which is the reciprocal of the exponential decay rate, satisfies
In this case, the decay estimate will be weaker: there is no exponential energy decay. In our opinion, this is quite natural due to the lack of coupling effects.
Moreover, concerning the influence of the parameter \(\beta \) on the decay rate, we also have
as for the case of a single degenerate wave equation (see [37]). Consequently, if \(\beta =0\), then exponential stability of the system (1) is still an open problem.
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References
Ammari, K., Tucsnak, M.: Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var. 6, 361–386 (2001)
Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992)
Bardos, C., Lebeau, G., Rauch, J.: Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problemes hyperboliques. Rend. Sem. Mat. Univ. Politec. Torino 46, 11–31 (1988)
Chen, G.: A note on the boundary stabilization of the wave equation. SIAM J. Control Optim. 19, 106–113 (1981)
Cox, S., Zuazua, E.: The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44, 545–573 (1995)
Gugat, M., Sigalotti, M., Tucsnak, M.: Robustness analysis for the boundary control of the string equation. In: Proceedings of the 9th European Control Conference, Kos, Greece (2007)
Haraux, A.: Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port. Math. 46, 245–258 (1989)
Haraux, A., Zuazua, E.: Decay estimates for some semilinear damped hyperbolic problems. Arch. Ration. Mech. Anal. 100, 191–206 (1988)
Komornik, V.: Exact Controllability and Stabilization (the Multiplier Method). Wiley, Paris (1995)
Lebeau, G., Robbiano, L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997)
Lions, J.L.: Contrôlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués, Tome 1. Masson, Paris (1988)
Lions, J.L.: Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30, 1–68 (1988)
Nicaise, S.: Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications. Rend. Mat. Appl. 23, 83–116 (2003)
Rauch, J., Taylor, M., Phillips, R.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974)
Tébou, L.R.T.: Stabilization of the wave equation with localized nonlinear damping. J. Differ. Equ. 145, 502–524 (1998)
Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15, 205–235 (1990)
Russell, D.L.: A general framework for the study of indirect damping mechanisms in elastic systems. J. Math. Anal. Appl. 173, 339–358 (1993)
Alabau-Boussouira, F.: Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM J. Control Optim. 41, 511–541 (2002)
Alabau-Boussouira, F., Léautaud, M.: Indirect stabilization of locally coupled wave-type systems. ESAIM Control Optim. Calc. Var. 18, 548–582 (2012)
Alabau-Boussouira, F., Cannarsa, P., Komornik, V.: Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2, 127–150 (2002)
Khodja, F.A., Bader, A.: Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force. SIAM J. Control Optim. 39, 1833–1851 (2001)
Akil, M., Ghader, M., Wehbe, A.: The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization. SeMA J. 78, 287–333 (2021)
Akil, M., Wehbe, A.: Indirect stability of a multidimensional coupled wave equations with one locally boundary fractional damping. Math. Nachr. 295, 2272–2300 (2022)
Akil, M., Badawi, H., Nicaise, S., Régnier, V.: Stabilization of coupled wave equations with viscous damping on cylindrical and non-regular domains: cases without the geometric control condition. Mediterr. J. Math. 19, 271 (2022)
Akil, M., Badawi, H., Nicaise, S.: Stability results of locally coupled wave equations with local Kelvin–Voigt damping: cases when the supports of damping and coupling coefficients are disjoint. Comput. Appl. Math. 41, 240 (2022)
Akil, M., Badawi, H., Nicaise, S., Wehbe, A.: Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface. Math. Methods Appl. Sci. 44, 6950–6981 (2021)
Wehbe, A., Issa, I., Akil, M.: Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients. Acta Appl. Math. 171, 1–46 (2021)
Akil, M., Chitour, Y., Ghader, M., Wehbe, A.: Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary. Asymptot. Anal. 119, 221–280 (2020)
Ammari, K., Mehrenberger, M.: Stabilization of coupled systems. Acta Math. Hungar. 123, 1–10 (2009)
Salah, M.B.H.: Stabilization of weakly coupled wave equations through a density term. Eur. J. Control 58, 315–326 (2021)
Allouni, H., Kesri, M., Benaissa, A.: On the asymptotic behaviour of two coupled strings through a fractional joint damper. Rendiconti del Circolo Matematico di Palermo Series 2(69), 613–640 (2020)
Gerbi, S., Kassem, C., Mortada, A., Wehbe, A.: Exact controllability and stabilization of locally coupled wave equations: theoretical results. Z. Anal. Anwend. 40, 67–96 (2021)
Kassem, C., Mortada, A., Toufayli, L., Wehbe, A.: Local indirect stabilization of n-d system of two coupled wave equations under geometric conditions. C. R. Math. 357, 494–512 (2019)
Liu, Z., Rao, B.: Frequency domain approach for the polynomial stability of a system of partially damped wave equations. J. Math. Anal. Appl. 335, 860–881 (2007)
Alabau-Boussouira, F.: On some recent advances on stabilization for hyperbolic equations. In: Control of Partial Differential Equations, pp. 1–100 (2012)
Bastin, G., Coron, J.M.: Stability and Boundary Stabilization of 1-d Hyperbolic Systems. Birkhäuser, Cham (2016)
Alabau-Boussouira, F., Cannarsa, P., Leugering, G.: Control and stabilization of degenerate wave equations. SIAM J. Control Optim. 55, 2052–2087 (2017)
Alabau-Boussouira, F., Wang, Z., Yu, L.: A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities. ESAIM Control Optim. Calc. Var. 23, 721–749 (2017)
Barbu, V.: Partial Differential Equations and Boundary Value Problems. Kluwer Academic Publishers, Dordrecht (1998)
Coron, J.M.: Control and Nonlinearity. American Mathematical Society, Providence (2007)
Lagnese, J.E.: Boundary Stabilization of Thin Plates. SIAM Studies in Applied Mathematics, Philadelphia (1989)
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Salhi, J., Moumni, A. & Tilioua, M. Indirect boundary stabilization of strongly coupled degenerate hyperbolic systems. Rend. Circ. Mat. Palermo, II. Ser 73, 1567–1590 (2024). https://doi.org/10.1007/s12215-024-01000-y
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DOI: https://doi.org/10.1007/s12215-024-01000-y