Neumann-Boundary Stabilization of the Wave Equation with Damping Control and Applications

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1 Neumann-Boundary Stabilization of the Wave Equation with Damping Control and Applications Boumediène CHENTOUF Aissa GUESMIA Abstract This article is devoted to boundary stabilization of a non-homogeneous n-dimensional wave equation subject to static or dynamic Neumann boundary conditions. Using a linear feedback law involving only a damping term, we provide a simple method and obtain an asymptotic convergence result for the solutions of the considered systems. The method consists in proposing a new energy norm. Then, a similar result is derived for the case of nonlinear damping feedback laws. Finally, the method presented in this work is also applied to a large class of distributed parameter systems such as the Petrovsky system, coupled wave-wave equations and elasticity systems. AMS MOS) Subject Classification. 35L05, 93C20, 93D15, 93D20 1 Introduction Let be a bounded open connected set in R n having a smooth boundary Γ = of class C 2. Given a partition Γ 0, ) of Γ, consider the following wave equation: y tt x, t) Ayx, t) = 0, in 0, ) 1.1) Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36. Al Khodh 123, Muscat, Sultanate of Oman. chentouf@squ.edu.om Metz Laboratory of Mathematics and Applications, University Paul Verlaine-Metz Metz, France. guesmia@univ-metz.fr This work was supported by Sultan Qaboos University Grant IG/SCI/DOMS/08/02). 1

2 with either static Neumann boundary conditions and initial conditions A yx, t) = 0, on Γ 0 0, ) A yx, t) = Ut), on 0, ) yx, 0) = y 0 x) H 1 ), y t x, 0) = z 0 x) L 2 ), or dynamical Neumann boundary conditions and initial conditions mx)y tt x, t) + A yx, t) = 0, on Γ 0 0, ) Mx)y tt x, t) + A yx, t) = Ut), on 0, ) yx, 0) = y 0 x) H 1 ), y t x, 0) = z 0 x) L 2 ), y t Γ0 x, 0) = w0x) 0 L 2 Γ 0 ), y t Γ1 x, 0) = w1x) 0 L 2 ), 1.2) 1.3) where A = n i,j=1 ia ij j ), A = n i,j=1 a ijν j j, k = x k, ν = ν 1,, ν n ) is the unit normal of Γ pointing towards the exterior of and a ij C 1 ) such that there exists α 0 > 0 satisfying a ij = a ji, i, j = 1,, n, a ij ɛ i ɛ j α 0 i,j=1 ɛ 2 i, ɛ 1,, ɛ n ) R n. 1.4) i=1 Moreover m L Γ 0 ); mx) m 0 > 0, x Γ 0 ; M L ); Mx) M 1 > 0, x. Furthermore, U is a feedback law depending only on a damping term, that is, 1.5) Ut) = ax) y t x, t), x, t) 0, ) 1.6) where the function a satisfies a L ); ax) a 0 > 0, x. 1.7) Note that is supposed to be nonempty whereas Γ 0 may be empty. In this article, it is proved that the solutions of each of the above closed loop system asymptotically tend towards a constant depending on the initial data y 0 and z 0. There is a rich literature concerning the stabilization problem of the wave equation with static boundary conditions 1.1)-1.2) and 1.6) see [2], [5]-[8], [26], [28]-[31], [33], [35]-[38] and the references therein). In all references cited above, at least one of the following conditions is assumed to be satisfied: 2

3 the equation 1.1) involves also the displacement term y. the stabilizing feedback law Ut) contains not only boundary dissipation y t but also a boundary displacement y. the first boundary condition in 1.2) involves the displacement term y the boundary condition 1.2) is replaced for instance by y = 0 or y ν + y = 0 on Γ 0 0, )). In other words, the term y is present in the closed-loop system. This type of assumption was impossible to circumvent for the stabilization of the wave equation because most of the authors defined the energy-norm of the system on the state space H 1 ) L 2 ) as: E 0 t) = 1 ) a ij i y j y + y t 2 dx, 2 i,j=1 which is only a semi-norm in our case. Note that J. Lagnese [27] has proved the energy decay of E 0 t) in the case when Γ 0 = and a ij = δ ij for the closed-loop system 1.1)-1.2) and 1.6). Nevertheless, the proof of this result is very technical and requires a preliminary result see Theorem 2 in [27]). We also point out that in the case when a ij = δ ij, m = 0 and M = 1, it was shown in [39] and [40] that the closed loop system 1.1), 1.3) and 1.6) is strongly stable but under two restrictive conditions: the domain is a bounded open set in R 2, n = 2). y = 0 on a part of the boundary Γ =. The main contribution of this paper is twofolds: i) to provide an alternative proof of Lagnese s result [27] by means of a simple and direct method and ii) to extend the results of [13], where the one-dimensional equation is dealt with, and those of [39] and [40] to a general domain in R n and a general operator A, and also without any boundary displacement term y in the system. The key idea of the proof is to introduce the following new energy associated to the closed-loop system 1.1)-1.2) and 1.6): { ) Et) = 1 a ij i y j y + y t 2 dx 2 i,j=1 [ ] } 2 + ɛ y t dx + ay dσ, 1.8) where ɛ > 0. Note that Et) is a perturbation of the energy E 0 t) used in literature. The choice of the new energy Et) is justified by the fact that the usual energy function E 0 t) 3

4 only induces a semi-norm on the state space H 1 ) L 2 ) for the closed-loop system 1.1)-1.2) and 1.6). In fact, the constants are solutions of 1.1)-1.2) and 1.6) and have null energy for E 0 t). An easy formal computation shows that Ėt) = a y t 2 dσ 0, 1.9) and thus the energy Et) is non-increasing. Concerning the closed-loop system 1.1), 1.3) and 1.6), the following energy is considered: { ) E d t) = 1 a ij i y j y + y t 2 dx + m y t 2 dσ + M y t 2 dσ 2 i,j=1 Γ 0 [ + µ y t dx + my t dσ + Myt + ay ) ] } 2 dσ, 1.10) Γ 0 where µ > 0. An easy formal computation shows that Ė d t) = a y t 2 dσ 0, 1.11) and thus the energy E d t) is non-increasing. The main results of this article are: i) For any initial data y 0, z 0 ) H 1 ) L 2 ), the solutions of the closed-loop system 1.1)-1.2) and 1.6) satisfy: yt), y t t)) C, 0) in H 1 ) L 2 ) as t +, where ) 1 { } C = a dσ z 0 dx + ay 0 dσ. 1.12) ii) The solutions of the closed-loop system 1.1), 1.3) and 1.6) stemmed from any initial condition y 0, z 0, w 0 0, w 0 1) H 1 ) L 2 ) L 2 Γ 0 ) L 2 ) satisfy: where C is defined by 1.12). yt), y t t), y t Γ0 t), y t Γ1 t)) C, 0, 0, 0) as t +, The nonlinear case of boundary control is also treated in this paper which is organized as follows. In the next section, the well-posedness and the asymptotic convergence of solutions for the wave-static boundary conditions system 1.1)-1.2) and 1.6) are established. Section 3 is dedicated to the wave equation with dynamical boundary conditions. More precisely, we shall prove that such a system is well-posed in the sense of semigroups theory and its solutions converge asymptotically to a constant for both linear and nonlinear damping controls. Some of the results of these two sections have been partially 4

5 announced in [11, 12] for simpler systems without detailed proofs. Section 4 is devoted to applications for several systems such as Petrovsky system, coupled wave-wave equations and elasticity systems. Finally, in the last section, some open problems are discussed. 2 The wave equation with static boundary conditions 2.1 Preliminaries and well-posedness of the problem In this subsection, we study the existence and uniqueness of the solutions of the closedloop system 1.1)-1.2) and 1.6). Consider the state space H = H 1 ) L 2 ), equipped with the inner product ) y, z), ỹ, z) H = a ij i y j ỹ + z z dx i,j=1 ) + ɛ z dx + ay dσ ) z dx + aỹ dσ, 2.1) where ɛ > 0 is a constant to be determined. The first result is stated in the following proposition: Proposition 2.1. The state space H = H 1 ) L 2 ) endoweded with the inner product 2.1) is a Hilbert space provided that ɛ is small enough. Proof of Proposition 2.1. It suffices to show that the norm H induced by the inner product 2.1) is equivalent to the usual one H 1 ) L 2 ), that is, the existence of two positive constants K and K such that K y, z) H 1 ) L 2 ) y, z) H K y, z) H 1 ) L 2 ). 2.2) On one hand, applying Hölder s inequality and using trace Theorem [1] see also [32]) and noting that there exists α 1 > 0 such that a ij ɛ i ɛ j α 1 ɛ 2 i, ɛ 1,, ɛ n ) R n, 2.3) yields: y, z) 2 H i,j=1 i=1 α1 y 2 + [1 + 2ɛ vol)] z 2) dx + 2ɛ a 2 C 1 vol ) y ɛ) y 2) ) dx, 5

6 where u = u L and C 1 is a positive constant depending on see [1] or [32]). Therefore the first inequality in 2.2) holds for a positive constant K depending on α 1, ɛ, a, vol ) and vol). For the reverse inequality, we proceed as follows: ) 2 ) 2 y, z) 2 H = a ij i y j y + z 2 dx + ɛ z dx) + ɛ ay dσ i,j=1 ) ) + 2ɛ ay dσ z dx. 2.4) Obviously for any δ > 0, Young s inequality gives ) ) 2 ) 2 2 ay dσ z dx δ 1 z dx) δ ay dσ. 2.5) Combining 1.4), 2.4) and 2.5), we get y, z) 2 H α0 y 2 + z 2) 2 ) 2 dx+ɛ1 δ 1 ) z dx) +ɛ1 δ) ay dσ. 2.6) Using a classical compactness argument, one can show the following generalized Poincaré inequality: { ) } 2 y 2 dx C 2 y 2 dx + y dσ, where C 2 > 0 depends on. This, together with 1.7), implies that ) 2 ay dσ a2 0 C 2 y 2 dx a 2 0 y 2 dx. 2.7) Inserting 2.6) into 2.7), it follows that { [ɛa ] y, z) 2 2 H 01 δ)c2 1 y 2 + [ α 0 ɛa 2 01 δ) ] } y 2 + z 2 dx 2 + ɛδ 1)δ 1 z dx), 2.8) for any 0 < δ < 1. Finally, applying again Hölder s inequality to the last term, one can show the existence of a positive constant K depending on α 0, a, δ and vol) such that the second inequality in 2.2) holds, provided that 0 < δ < 1 and ɛ satisfies the following condition ɛ < min α 0 a 2 01 δ), This concludes the proof of Proposition 2.1. ) δ. 1 δ)vol) 6

7 We turn now to the formulation of the closed-loop system 1.1)-1.2) and 1.6) in an abstract form in H. Let zt) = y t t) and Φt) = yt), zt)). Then, the closed loop system can be written as follows { Φt t) = AΦt), Φ0) = Φ 0 = y 0, z 0 ), where A is an unbounded linear operator defined by: } DA) = {y, z) H 1 ) H 1 ); Ay L 2 ); A y = 0 on Γ 0 ; A y + az = 0 on, and for any y, z) DA), 2.9) 2.10) Ay, z) = z, Ay). 2.11) Now we are able to state a well-posedness result for the closed-loop system 2.9): Lemma 2.1. i) The linear operator A, defined by 2.10)-2.11), generates a C 0 semigroup of contractions St) on H = DA). ii) For any initial data Φ 0 = y 0, z 0 ) DA), the system 2.9) admits a unique strong solution Φt) = yt), y t t)) = St)Φ 0 DA) for all t 0 such that Φ = y, y t ) C 1 R + ; H) CR + ; DA)). Moreover, the function t AΦt) H is non-increasing. iii) For any initial data Φ 0 = y 0, z 0 ) H, the system 2.9) has a unique weak solution Φt) = yt), y t t)) = St)Φ 0 H such that Φ = y, y t ) CR + ; H). Proof of Lemma 2.1. i) Let φ = y, z) DA). Using Green formula, one can obtain after a straightforward computation < Ay, z), y, z) > H = a z 2 dσ ) Therefore A is monotone. Moreover, using Lax-Milgram Theorem [4], one can prove that A is maximal, that is, RI A) = H. Thus, Lummer-Phillips Theorem [34] leads us to claim that A generates a C 0 semigroup of contractions St) on H = DA). ii)-iii) These claims are direct consequences of semigroup theory [34]. 2.2 Asymptotic behavior In this subsection, we will show an asymptotic behavior result for the unique solution of the system 2.9) in H. To do so, we shall first show the following lemma: Lemma 2.2. The resolvent operator λi A) 1 : H H is compact for any λ > 0 and hence the canonical embedding i : DA) H is compact, where DA) is equipped with the graph norm. 7

8 Proof of Lemma 2.2. Based on the proof of Lemma 2.1 and using Sobolev embedding, one deduces that I A) 1 is compact. The proof of Lemma 2.2 follows then from the well-known result of Kato [24]. The first main result of this paper is: Theorem 2.1. For any initial data Φ 0 = y 0, z 0 ) H, the solution Φt) = yt), y t t)) of 2.9) tends in H to C, 0) as t +, where ) 1 ) C = a dσ z 0 dx + ay 0 dσ. Proof of Theorem 2.1. By a standard argument of density of DA 2 ) in H and the contraction of the semigroup St), it suffices to prove Theorem 2.1 for smooth initial data Φ 0 = y 0, z 0 ) DA 2 ). Let Φt) = yt), y t t)) = St)Φ 0 be the solution of 2.9). It follows from Lemma 2.1 ii) that the trajectory of solution {Φt)} t 0 is a bounded set for the graph norm and thus precompact by virtue of Lemma 2.2. Applying LaSalle s principle, we deduce that ω Φ 0 ) is non empty, compact, invariant under the semigroup St) and in addition St)Φ 0 ω Φ 0 ) as t + [23]. In order to prove the result, it suffices to show that ω Φ 0 ) reduces to C, 0). To this end, let Φ 0 = ỹ 0, z 0 ) ω Φ 0 ) DA) and Φt) = ỹt), ỹ t t)) = St) Φ 0 DA) the unique strong solution of 2.9). Recall that it is well-known that Φt) H is constant [23] and thus dt d ) Φt) 2 H = 0, i.e, < A Φ, Φ > H = a z 2 dσ = ) This implies that z = ỹ t = 0 on and therefore ỹ is solution of the following system: ỹ tt Aỹ = 0, in A ỹ = 0, on Γ 0 ỹ t = A ỹ = 0, on 2.14) ỹ0) = ỹ 0 ; ỹ t 0) = z 0, ỹ H 1 ); Aỹ L 2 ), in and z = ỹ t is solution of z tt A z = 0, in A z = 0, on Γ 0 z = A z = 0, on z0) = z 0, z t 0) = Aỹ 0, in. 2.15) 8

9 Obviously, to deduce the desired result, it suffices to show that ỹ = constant is the only solution of 2.14). To do so, we first use the standard Holmgren s uniqueness theorem for the system 2.15) to conclude that z 0. Thus the system 2.14) is reduced to an elliptic problem: Aỹ = 0, in A ỹ = 0, on Γ 0 A ỹ = 0, on, which clearly yields that ỹ constant. Thus, we have proved that for any Φ 0 = ỹ 0, z 0 ) ω Φ 0 ) DA), the solution Φt) = ỹt), ỹ t t)) = St) Φ 0 DA) satisfies ỹt), ỹ t t)) = C, 0), for any t 0, where C is a constant. In particular, Φ0 = ỹ 0, z 0 ) = C, 0) and hence the ω-limit set ω Φ 0 ) consists of constants C, 0). Now, we are able to find C. Let C, 0) ω Φ 0 ). This implies that there exists {t n }, as n such that Φt n ) = yt n ), y t t n )) = St n )Φ 0 C, 0) 2.16) in the state space H = H 1 ) L 2 ). Furthermore, any solution of the closed-loop system 1.1)-1.2) and 1.6) stemmed from Φ 0 = y 0, z 0 ) verifies { } d y t x, t) dx + ayx, t) dσ = 0. dt This can be obtained by integrating y tt x, t) Ayx, t) = 0 with respect to x, then using Green formula and finally using the boundary conditions 1.2) and 1.6)). Hence y tx, t) dx + ayx, t) dσ is constant and so y t x, t) dx + ayx, t) dσ = y t x, 0) dx + ayx, 0) dσ Γ 1 = z 0 x) dx + ay 0 x) dσ. 2.17) Finally, let t = t n in 2.17) with n and use 2.16) to obtain: ) 1 ) C = a dσ z 0 dx + ay 0 dσ. This achieves the proof of the theorem. 3 The wave equation with dynamical boundary conditions In this section, we treat the case when the boundary conditions are dynamical. Indeed, we shall consider both linear and nonlinear damping control. 9

10 3.1 Dynamical boundary conditions with a linear damping control Well-posedness of the problem In this subsection, we study the well-posedness of the problem 1.1)-1.3) and 1.6). To do so, let us consider the state space H d = H 1 ) L 2 ) L 2 Γ 0 ) L 2 ), equipped with the inner product ) y, z, w 0, w 1 ), ỹ, z, w 0, w 1 ) = a ij i y j ỹ + z z dx + mw 0 w 0 dσ+ H d i,j=1 Γ 0 Mw 1 w 1 dσ ) +µ z dx + mw 0 dσ + Mw 1 + ay) dσ z dx + m w 0 dσ Γ 0 Γ 0 ) + M w 1 + aỹ) dσ, where µ > 0 is a constant to be determined. We have following proposition: 3.1) Proposition 3.1. The state space H d equipped with the inner product 3.1) is a Hilbert space provided that µ is small enough. Proof of Proposition 3.1. It suffices to show that the norm Hd induced by the inner product 3.1) is equivalent to the usual one denoted by, that is, prove the existence of two positive constants K and K such that K y, z, w 0, w 1 ) y, z, w 0, w 1 ) Hd K y, z, w 0, w 1 ). 3.2) On one hand, using 1.5) and applying Cauchy-Schwartz inequality yields: ) y, z, w 0, w 1 ) 2 H d a ij i y j y + z 2 dx + m w 0 Γ 2 dσ 0 i,j=1 2 ) 2 + M w 1 Γ 2 dσ + 4µ z dx) + 4µ mw 0 dσ 1 Γ 0 ) 2 ) 2 + 4µ Mw 1 dσ + 4µ ay dσ. 3.3) 10

11 Then we apply Hölder s inequality for the four last terms of the right-hand side of 3.3) and use 1.5) and 1.7) to get ] y, z, w 0, w 1 ) 2 H d [α 1 y µ vol)) z 2 dx ) + C m w 0 Γ 2 dσ + C M w 1 2 dσ + 4µ a 2 vol ) y 2 dσ, 0 where C m = m [1 + 4µ m volγ 0 )], C M defined in 2.3). = M [1 + 4µ M vol )] and α 1 is On the other hand, using trace Theorem [32], the above inequality becomes y, z, w 0, w 1 ) 2 H d 4µ a 2 C 1 vol ) y 2 dx + α 1 + 4µ a 2 C 1 vol ) ) y 2 dx µ vol)) z 2 dx + C m w 0 Γ 2 dσ + C M w 1 2 dσ, 0 where C 1 is a positive constant depending on see [32]). Therefore y, z) 2 H d K y, z) 2 H 1 ) L 2 ), where K is a positive constant depending on α 1, µ, a, m, M, volγ i ), i = 0, 1 and vol). For the reverse inequality, we proceed as follows: ) y, z, w 0, w 1 ) 2 H d = a ij i y j y + z 2 dx + m w 0 2 dσ + M w 1 2 dσ i,j=1 Γ 0 [ ] 2 [ ] 2 + µ z dx + mw 0 dσ + Mw 1 + ay) dσ + µ ay dσ Γ 0 Γ [ ] [ 1 ] + 2µ ay dσ z dx + mw 0 dσ + Mw 1 + ay) dσ.3.4) Γ 0 Applying Cauchy-Schwartz inequality yields ) ) 2 ay dσ z dx + mw 0 dσ + Mw 1 + ay) dσ Γ 0 Γ 1 ) 2 ) 2 δ 1 z dx + mw 0 dσ + Mw 1 + ay) dσ δ ay dσ, Γ 0 3.5) for any δ > 0. Now, using 1.4), 1.5) and combining 3.4) and 3.5), we get y, z, w 0, w 1 ) 2 H d α0 y 2 + z 2) dx + m 0 w 0 2 dσ + M 1 w 1 2 dσ Γ 0 +µ 1 δ 1) ) 2 ) 2 z dx + mw 0 dσ + Mw 1 + ay) dσ + µ1 δ) ay dσ. Γ 0 3.6) 11

12 Choosing 0 < δ < 1 and applying Cauchy-Schwartz and Hölder s inequalities for the term z dx + mw Γ0 0 dσ + ) 2 Mw Γ1 1 + ay) dσ of the right-hand side of 3.6), we obtain y, z, w 0, w 1 ) 2 H d α 0 y 2 dx + 1 3µ1 δ)δ 1 vol) ) z 2 dx + m 0 1 3µm0 1 δ)δ 1 volγ 0 ) ) w 0 2 dσ 3.7) Γ 0 Inserting 2.7) into 3.7) yields + M 1 1 3µM1 1 δ)δ 1 vol ) ) w 1 2 dσ ) 2 + µ1 δ) ay dσ. y, z, w 0, w 1 ) 2 H d µ1 δ)a 2 0C µ1 δ)δ 1 vol) ) z 2 dx y 2 dx + ) α 0 µ1 δ)a 2 0 y 2 dx + m 0 1 3µm0 1 δ)δ 1 volγ 0 ) ) Γ 0 w 0 2 dσ + M 1 1 3µM1 1 δ)δ 1 vol ) ) w 1 2 dσ, for any 0 < δ < 1. Finally, provided that µ satisfies the following condition ) α 0 δ µ < min, 1 δ)a δ)vol), δ 31 δ)m 0 volγ 0 ), δ, 31 δ)m 1 vol ) one can provide a positive constant K depending on α 0, m 0, a 0, M 1, δ, C 2, vol) and volγ i ), i = 0, 1 such that the left inequality in 3.2) holds. This concludes the proof of Proposition 3.1. We turn now to the formulation of the closed-loop system 1.1)-1.3) and 1.6) in an abstract form on H d. Setting z = y t, w 0 = z Γ0, w 1 = z Γ1 and Φt) = yt), zt), w 0 t), w 1 t)), the closed loop system can be written into the following form: { Φt t) = A d Φt), Φ0) = Φ 0 = y 0, z 0, w 0 0, w 0 1), where A d is an unbounded linear operator defined by: 3.8) DA d ) = {y, z, w 0, w 1 ) H 1 ) H 1 ) L 2 Γ 0 ) L 2 ); Ay L 2 ), w 0 = z Γ0, w 1 = z Γ1 }, 3.9) 12

13 and for any y, z, w 0, w 1 ) DA d ), ) A d y, z, w 0, w 1 ) = z, Ay, 1 m Ay, 1 M aw 1 + A y). 3.10) Now we are able to state a well-posedness result for the closed-loop system 3.8): Lemma 3.1. i) The linear operator A d, defined by 3.9)-3.10), generates a C 0 semigroup of contractions S d t) on H d = DA d ). ii) For any initial data Φ 0 = y 0, z 0, w0, 0 w1) 0 DA d ), the system 3.8) admits a unique strong solution Φt) = yt), y t t), w 0 t), w 1 t)) = S d t)φ 0 DA d ) for all t 0 such that Φ C 1 R + ; H d ) CR + ; DA d )). Moreover, the function t A d Φt) Hd is nonincreasing. iii) For any initial data Φ 0 = y 0, z 0, w0, 0 w1) 0 H d, the system 3.8) has a unique weak solution Φt) = yt), y t t), w 0 t), w 1 t)) = S d t)φ 0 H d such that Φ CR + ; H d ). Proof of Lemma 3.1. i) Let φ = y, z, w 0, w 1 ) DA d ). Using Green formula, one can obtain after a straightforward computation < A d y, z, w 0, w 1 ), y, z, w 0, w 1 ) > Hd = a w 1 2 dσ ) Therefore A d is monotone. Now, given f, g, ξ, η) H d, we seek y, z, w 0, w 1 ) DA d ) solution of the equation I A d )y, z, w 0, w 1 ) = f, g, ξ, η), that is, y z = f, in z Ay = g, in w m Ay = ξ, on Γ 0, w M aw 1 + A y) = η, on. Then eliminating z and using w 0 = z Γ0, w 1 = z Γ1, we find that y satisfies the system Ay y = f + g) L 2 ), in y + 1 m Ay = ξ, on Γ ) 1 + a ) y + 1 M M Ay = 1 + a ) f + η, on. M Using Green formula, one can prove that the system 3.12) is equivalent to the following variational equation ) yψ + a ij i y j ψ dx + myψ dσ + M + a)yψ dσ i,j=1 Γ 0 Γ 1 = f + g) ψ dx + mξψ dσ + Mη + M + a)f) ψ dσ, Γ )

14 for any ψ H 1 ). Thanks to Lax-Milgram Theorem [4], one can prove that 3.13) admits a unique solution y H 1 ). Then defining z = y f, w 0 = z Γ0 and w 1 = z Γ1, we find that the element y, z, w 0, w 1 ) DA d ) and is solution of the equation I A d )y, z, w 0, w 1 ) = f, g, ξ, η). Thus A is maximal, that is, RI A d ) = H d. Finally, Lummer-Phillips theorem [34] permits us to claim that A d generates a C 0 semigroup of contractions S d t) on H d = DA d ). ii)-iii) These claims follow from semigroups theory [34] Asymptotic behavior for the solution of 3.8) In this subsection, we will show an asymptotic behavior result for the unique solution of 3.8) in H d. To do so, one can use Lemma 3.1 and Sobolev embedding to show that the canonical embedding i : DA d ) H d is compact, where DA d ) is equipped with the graph norm. Therefore I A d ) 1 isolated eigenvalues with finite multiplicity [24]. The second main result of this paper is: is compact and the spectrum of A d consists of only Theorem 3.1. For any initial data Φ 0 = y 0, z 0, w 0 0, w 0 1) H d, the solution Φt) = yt), y t t), w 0 t), w 1 t)) of 3.8) tends in H d to C, 0, 0, 0) as t +, where ) 1 { } C = a dσ z 0 dx + ay 0 dσ. Proof of Theorem 3.1. By a standard argument of density of DA 2 d ) in H d and the contraction of the semigroup S d t), it suffices to prove Theorem 3.1 for smooth initial data Φ 0 = y 0, z 0, w 0 0, w 0 1) DA 2 ). Let Φt) = yt), y t t), w 0 t), w 1 t)) = S d t)φ 0 be the solution of 3.8). It follows from Lemma 3.1 ii) that the trajectory of solution {Φt)} t 0 is a bounded set for the graph norm and thus precompact by virtue of the compactness of the operator I A d ) 1. Applying LaSalle s principle, we deduce that ω Φ 0 ) is non empty, compact, invariant under the semigroup S d t) and in addition S d t)φ 0 ω Φ 0 ) as t + [23]. In order to prove the strong stability, it suffices to show that ω Φ 0 ) reduces to C, 0, 0, 0). To this end, let Φ 0 = ỹ 0, z 0, w 0, w 1 ) ω Φ 0 ) DA d ) and Φt) = ỹt), ỹ t t), w 0 t), w 1 t)) = S d t) Φ 0 DA d ) the unique strong solution of 3.8). Recall that it is well-known that Φt) Hd = 2E d t) see 1.10)) is constant [23] and thus dt d ) Φt) 2Hd = 0, i.e, E d t) =< A d Φ, Φ >Hd = ) 14

15 This, together with 3.11) see also 1.11)), implies that w 1 = y t Γ1 = 0 and therefore ỹ is solution of the following system: ỹ tt Aỹ = 0, in mỹ tt + A ỹ = 0, on Γ 0, ỹ t = A ỹ = 0, on, 3.15) ỹ0) = ỹ 0 ; ỹ t 0) = z 0, in ỹ H 1 ); Aỹ L 2 ). A straightforward computation shows that z = ỹ t is solution of z tt A z = 0, in z = A z = 0, on Γ 0, z = A z = 0, on, z0) = z 0 ; z t 0) = Aỹ 0. in. 3.16) Obviously, to deduce the desired result, it suffices to show that ỹ = constant is the only solution of 3.15). To do so, we first use the standard Holmgren s uniqueness theorem for the system 3.16) to conclude that z = 0. Thus the system 3.15) is reduced to an elliptic problem: Aỹ = 0, in A ỹ = 0, on Γ 0, A ỹ = 0, on, which clearly yields that ỹ is constant. This, together with 3.14), implies the desired result in the same way as for the proof of Theorem 3.1. Remark 3.1. Integrating with respect to x and t and using Green formula for the closed loop system 1.1)-1.2) and 1.6) resp. 1.1), 1.3) and 1.6)), we obtain the following identity: y t dx + ay dσ = z 0 dx + ay 0 dσ. Therefore if the initial values y 0 and z 0 ) satisfy the additional condition z 0 dx + ay 0 dσ = 0 then the constant C of Theorem 2.1 resp. Theorem 3.1) is zero. In other words, the energy defined by 1.8) resp. 1.10)) tends to 0 as t +. 15

16 3.2 Dynamical boundary conditions with a nonlinear damping control The aim of this section is to extend the results of the previous subsection to the case of nonlinear feedback control. For sake of simplicity and without loss of generality, we shall consider the system with constant coefficients y tt x, t) yx, t) = 0, in 0, ) my tt x, t) + y ν x, t) = 0, on Γ 0 0, ) My tt x, t) + y ν x, t) = fy tx, t)), on 0, ) yx, 0) = y 0 x) H 1 ), y t x, 0) = z 0 x) L 2 ), y t Γ0 x, 0) = w0x) 0 L 2 Γ 0 ), y t Γ1 x, 0) = w1x) 0 L 2 ), 3.17) where f satisfies the classical assumptions, namely, f is a non-decreasing continuous function such that f0) = 0. Then, one can check that, although with such a function f, the energy defined in 1.10) is not necessarily non-increasing. Hence a new methodology should be adopted. To do so, we shall consider, as in the previous subsection, our state space but equipped with the new inner product y, z, w 0, w 1 ), ỹ, z, w 0, w 1 ) = H d M w 1 w 1 dσ + ρ yỹ dσ, where ρ is any positive constant. H d = H 1 ) L 2 ) L 2 Γ 0 ) L 2 ), y ỹ + z z) dx + mw 0 w 0 dσ+ Γ ) Clearly, the norm induced by this inner product is equivalent to the usual one. Then one writes the system 3.17) in the space H d as follows: { Φt t) + B + P) Φt) = 0, 3.19) Φ0) = Φ 0 = y 0, z 0, w0, 0 w1), 0 where Φt) = yt), zt), w 0 t), w 1 t)), z = y t, w 0 = z Γ0, w 1 = z Γ1 operator defined by: and B is a nonlinear DB) = {y, z, w 0, w 1 ) H 1 ) H 1 ) L 2 Γ 0 ) L 2 ); y L 2 ), w 0 = z Γ0, w 1 = z Γ1 }, and for any y, z, w 0, w 1 ) DB), By, z, w 0, w 1 ) = z, y, 1 y m ν, 1 M fw 1 ) + y ) ) ν + ρy. 16

17 Moreover, P is a linear Lipschitz compact operator on H d such that Py, z, w 0, w 1 ) = 0, 0, 0, ρ M y). Our objective is to show that B is a nonlinear maximal monotone operator in H d [3]. To do so, a straightforward computation gives: By, z, w 0, w 1 ) Bỹ, z, w 0, w 1 ), y, z, w 0, w 1 ) ỹ, z, w 0, w 1 ) Hd = fw 1 ) f w 1 )) w 1 w 1 ) dσ 0, 3.20) for any φ = y, z, w 0, w 1 ), ỹ, z, w 0, w 1 ) DB). Thus, the operator B is monotone in H d. Next, we are going to show that B is maximal in H d. For this, given u, v, ξ, η) H d, we seek a solution y, z, w 0, w 1 ) DB) of the equation I + B)y, z, w 0, w 1 ) = u, v, ξ, η), i.e., with z = y u, in, w 0 = z Γ0 by where Jψ) = F ψ u) dσ y y = u + v, in y + 1 y m ν = u + ξ, on Γ 0, 1 + ρ ) y + 1 fy u) + y ) = u + η, on, M M ν ψ 2 + ψ) 2) dx + m 3.21) and w 1 = z Γ1. Now let us define the function J on H 1 ) ψ 2 dσ + ρ + M) ψ 2 dσ Γ 0 Γ 1 u + v)ψ dx m u + ξ)ψ dσ M u + η)ψ dσ, Γ 0 F x) = x 0 fs)ds, x R. From the assumptions on f, we deduce that J is convex, coercive and continuous in H 1 ). Hence by a minimization theorem [41], there exists a function y H 1 ) such that Jy) = inf Jψ). ψ H 1 ) This implies that the function Θ : λ Θλ) = Jy + λψ) admits a minimum at λ = 0 and thus d dλ Jy + λψ)) λ=0 = 0, ψ H 1 0, 1). This means that for any ψ H 1 0, 1), the element y is a weak solution of the system 3.21). Then one can show that y is indeed the unique solution see [9, 10] and the references 17

18 therein for similar arguments for one-dimensional systems). Therefore the operator B is maximal monotone on H d. This, together with the fact that P is a linear Lipschitz compact operator on H d, implies that the operator B + P) generates a C 0 -semigroup St) on H d see Remark 3.14, p. 106 in [3]). Now, using the semigroup theory of nonlinear operators see for instance [3]), one can claim that for any initial data y 0, z 0, w 0 0, w 0 1) DB), the system 3.17) or 3.19)) admits a unique strong solution y, z, w 0, w 1 ) = St)y 0, z 0, w 0 0, w 0 1) DB) such that d dt y, z, w 0, w 1 ) L R + ; H d ). In turn, for any initial data y 0, z 0, w 0 0, w 0 1) H d, the system 3.17) or 3.19)) has a unique weak y, z, w 0, w 1 ) = St)y 0, z 0, w 0 0, w 0 1) H d such that This yields in particular y, z, w 0, w 1 ) CR + ; H d ). 0 w 1 t)fw 1 t))dt <. 3.22) Integrating the equation 3.17) and assuming that the function f is differentiable, it follows that y t x, t)dx + m y t x, t)dσ + Γ 0 My t x, t) + f 0)yx, t)) dσ = +m y t x, 0)dσ + Γ 0 t My t x, 0) + f 0)yx, 0)) dσ + 0 f 0)y t x, s) fy t x, s))) dσ. y t x, 0)dx 3.23) Then, suppose that there exists a positive constant K such that f satisfies the following hypothesis: f 0)s fs) Ksfs), 3.24) for any s in some neighborhood of 0. This leads us to claim that the solution y, z, w 0, w 1 ) = y, y t, y t Γ0, y t Γ1 ) stemmed from any initial condition y 0, z 0, w 0 0, w 0 1) in H d satisfies the following: the function t y, y t, y t Γ0, y t Γ1 ) is bounded in H d. This, together with 3.22)-3.24), implies that as t, the solution yt), y t t), y t Γ0 t), y t Γ1 t)) C, 0, 0, 0 ), 18

19 where C = f 0)vol )) f 0)w 1 s) fw 1 s))) dσ. 4 Applications to other systems ) ) z 0 dx + m w0dσ 0 + Mw f 0)y 0) dσ Γ 0 The method presented in the previous sections can be applied for a large class of distributed systems where the classical energy defines only a semi-norm in the state space) to prove that the solution converges to an equilibrium point when the time goes to infinity) which can be determined. We give here some particular applications to Petrovsky system, coupled wave-wave equations and elasticity systems. There are many results concerning the stability of this type of systems see [14]-[22] and the references therein) with different controls linear, nonlinear, internal, boundary, of memory,...) and different boundary conditions Dirichlet, Neumann,...). In all these works, the considered contexts of systems guarantee that the classical energy defines a norm on the state space. This property is not valid in the case of our applications. 4.1 Petrovsky system Let be a bounded open connected set in R n having a smooth boundary Γ = of class C 4, with a partition Γ 0, ). Then, consider the following Petrovsky system: y tt x, t) + 2 yx, t) = 0, in 0, ) ν yx, t) = 0, on Γ 0, ) ν yx, t) = 0, on Γ 0 0, ) ν yx, t) = ay t x, t), on 0, ) yx, 0) = y 0 x), y t x, 0) = z 0 x), in where a is defined by 1.7). The classical energy is defined by Ẽ 0 t) = 1 y 2 + y t 2) dx, 2 which is only a semi-norm for y, y t ) in the space H p = H 2 ) L 2 ). The new energy associated to our system is defined by [ E p t) = Ẽ0t) + ɛ y t dx + ay dσ 19 ] 2 4.1)

20 which is a norm in H p equivalent to the usual one of H 2 ) L 2 ), for ɛ > 0 small enough. An easy formal computation shows that Ė p t) = a y t 2 dσ 0, and thus the energy E p t) is non-increasing. Using LaSalle s principle and following the arguments used before, we obtain that, for any initial data H p, the solutions of 4.1) satisfy: yt), y t t)) C, 0) in H p as t +, where ) 1 { } C = a dσ z 0 dx + ay 0 dσ. Remark 4.1. One can consider dynamical boundary conditions as in 1.3), that is ν yx, t) = 0, on Γ 0, ) mx)y tt x, t) + ν yx, t) = 0, on Γ 0 0, ) Mx)y tt x, t) + ν yx, t) = ay t x, t), on 0, ) yx, 0) = y 0 x) H 1 ), y t x, 0) = z 0 x) L 2 ), y t Γ0 x, 0) = w0x) 0 L 2 Γ 0 ), y t Γ1 x, 0) = w1x) 0 L 2 ), 4.2) where m and M are defined by 1.5). We obtain that yt), y t t), y t Γ0 t), y t Γ1 t)) C, 0, 0, 0) in H 2 ) L 2 ) L 2 Γ 0 ) L 2 ) as t + where C is defined above. Furthermore, one can propose on a nonlinear damping term fy t x, t)) for the system 4.2) and use the same arguments as in subsection 3.2 to obtain very similar results. 4.2 Coupled wave-wave equations Let be a bounded open connected set in R n having a smooth boundary Γ = of class C 2, with a partition Γ 0, ). Here we consider the following coupled system: y tt x, t) + Ayx, t) + bu tt = 0, in 0, ) u tt x, t) + Bux, t) + by tt = 0, in 0, ) A yx, t) = B ux, t) = 0, on Γ 0 0, ) A yx, t) = a 1 y t x, t), on 0, ) 4.3) B ux, t) = a 2 u t x, t), on 0, ) yx, 0) = y 0 x), y t x, 0) = z 0 x), in ux, 0) = u 0 x), u t x, 0) = v 0 x), in 20

21 where A = n i,j=1 ia ij j ), B = n i,j=1 ib ij j ), A = n i,j=1 a ijν j j, B = n i,j=1 b ijν j j, a ij, b ij C 1 ) such that there exists α 0, β 0 > 0 satisfying a ij = a ji, b ij = b ji, i, j = 1,, n, a ij ɛ i ɛ j α 0 ɛ 2 i, b ij ɛ i ɛ j β 0 ɛ 2 i, ɛ 1,, ɛ n ) R n. i,j=1 i=1 i,j=1 Moreover, a 1 and a 2 are defined as for the function a in 1.7), and b L ) satisfying b < 1. The classical energy is defined by Ê 0 t) = 1 ) a ij i y j y + b ij i u j u dx 2 i,j=1 i,j=1 + 1 ) yt 2 + u t 2 + 2by t u t dx 2 which is only a semi-norm for y, u, y t, u t ) in the space i=1 H w = H 1 )) 2 L 2 )) 2. The new energy associated to our system is defined by [ E w t) = Ê0t) + ɛ b + 1)y t + u t ) dx + a 1 y + a 2 u) dσ which is a norm in H w equivalent to the usual one of H 1 )) 2 L 2 )) 2 for ɛ > 0 small enough note that, thanks to the fact that b < 1, the expression ) yt 2 + u t 2 + 2by t u t dx defines a norm for y t, u t ) that is equivalent to that of L 2 )) 2 ). An easy formal computation shows that Ė w t) = a 1 y t 2 + a 2 u t 2 )dσ 0, and thus the energy E w t) is non-increasing. Using LaSalle s principle and following the arguments used before, we obtain that, for any initial data y 0, u 0, z 0, v 0 ) H w, the solutions of 4.3) satisfy: yt), ut), y t t), u t t)) C, C, 0, 0) in H w as t +, where ) ) { } C a 1 dσ + C a 2 dσ = 1 + b)z 0 + v 0 ) dx + a 1 y 0 + a 2 u 0 ) dσ. If A = B, a 1 = a 2 and y 0, z 0 ) = u 0, v 0 ), then, it follows from the symmetry that ) 1 { } C = C = a 1 dσ 1 + b)z 0 dx + a 1 y 0 dσ. 21 ] 2

22 Remark 4.2. i) We can consider dynamical boundary conditions as 1.3) and then obtain the same result with the same constants C and C as before. ii) We can consider Neumann or dynamical boundary conditions only for y, and the homogeneous Dirichlet one for u or the reverse). In this case, we get where yt), ut), y t t), u t t)) C, 0, 0, 0), ) 1 { } C = a 1 dσ 1 + b)z 0 dx + a 1 y 0 dσ. iii) We can obtain the same results if we consider coupled Petrovsky-Petrovsky or wave- Petrovsky system with Neumann or dynamical boundary conditions as in 1.2) or 1.3) for the wave equation, and as in 4.1) or 4.2) for Petrovsky one). iv) In the case of dynamical boundary conditions, it is easy to check that a nonlinear boundary damping term may be used to obtain similar results to those of subsection Elasticity systems Let be a bounded open connected set in R n having a smooth boundary Γ = of class C 2, with a partition Γ 0, ). Now let us consider the following elasticity system: y itt x, t) n j=1 σ ij,jx, t) = 0, in 0, ), i = 1,, n n j=1 σ ijν j = 0, on Γ 0 0, ), i = 1,, n n j=1 σ ijν j = a i y it x, t), on 0, ), i = 1,, n y i x, 0) = y 0 i x), y it x, 0) = z 0 i x), in, i = 1,, n 4.4) here y = y 1,, y n ) : R n is the solution, the a i are defined as a in 1.7), σ ij,j = σ ij x j, σ ij = n k,l=1 a ijklε kl, ε ij = 1y 2 i,j + y j,i ), y i,j = y i x j, y j,i = y j x i that there exists α 0 > 0 satisfying a ijkl = a klij = a jikl, i, j, k, l = 1,, n, i,j,k,l=1 a ijkl ɛ ij ɛ kl α 0 ɛ ij ɛ ij i,j=1 and a ijkl C 1 ) such for all symmetric tensor ɛ ij. For more details concerning these systems, see [14]-[17] and the references therein. The classical energy of this system is defined by Ē 0 t) = 1 ) σ ij ε ij + y it 2 dx, 2 i,j=1 22 i=1

23 which is only a semi-norm for y, y t )in the space H e = H 1 )) n L 2 )) n. The new energy associated to our system is defined by [ E e t) = Ē0t) + ɛ y it dx + i=1 a i y i dσ which is a norm in H e equivalent to the usual one of H 1 )) n L 2 )) n for ɛ > 0 small enought thanks to the Korn inequality). An easy formal computation shows that Ė e t) = a i y it 2 dσ 0, i=1 and thus the energy E e t) is non-increasing. Using LaSalle s principle and following the arguments used before, we obtain that, for any initial data y 0, z 0 ) H e, the solutions of 4.4) satisfy: yt), y t t)) C, 0) in H e as t +, where C = C 1,, C n ) and ) { } C i a i dσ = zi Γ 0 dx + a i yi 0 dσ. 1 i=1 i=1 If a i = a j, yi 0 = yj 0 and zi 0 = zj 0 for all i, j = 1,, n, then, by symmetry, we have ) 1 { } C i = a 1 dσ z1 0 dx + a 1 y1 0 dσ, for all i = 1, n. Remark 4.3. We can consider Neumann conditions for y i, i = 1,, r, dynamical boundary conditions for y i, i = r + 1,, p, and the homogeneous Dirichlet ones for y i, i = p + 1,, n, where 0 r p n; that is n j=1 σ ijν j = 0, on Γ 0 0, ), i = 1,, r i=1 i=1 ] 2 n j=1 σ ijν j = a i y it x, t), n j=1 σ ijν j + m i y itt = 0, on 0, ), i = 1,, r on Γ 0 0, ), i = r + 1,, p n j=1 σ ijν j + M i y itt = a i y it x, t), 4.5) y i = 0, on 0, ), i = r + 1,, p on Γ 0, ), i = p + 1,, n In this case, the energy of the system will tend to C, where C i = 0 for i = p + 1,, n, and p i=1 ) { C i a i dσ = zi 0 dx + i=1 } a i yi 0 dσ. Moreover, the method adopted in subsection 3.2 can also be used to treat the case when the damping boundary control is nonlinear. 23 i=1

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