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Local Indirect Stabilization of two coupled wave equations under geometric conditions
Let be a non empty open bounded domain of RN with boundary of class C2.
In chapter 2, we consider the following coupled wave equation :
utt a u + c(x)ut + b(x)yt = 0 in R+; (1.1.4)
ytty b(x)ut = 0 in R+; (1.1.5)
u = y = 0 on R+; (1.1.6)
with the following initial data :
u(x; 0) = u0; y(x; 0) = y0; ut(x; 0) = u1 and yt(x; 0) = y; x 2 ;
where a > 0 constant and b(x) 2 C0( ; R) is a non-zero function. The damping term c(x) 2 C 0( ; R+) is only applied at the rst equation and the second equation is indirectly damped through the coupling between the two equations. This type of indirect control was introduced by D.L. Russel [63] and since this time, it attracted the attention of many authors.
Preceding results :
In [37], B. Kapitonov studied the stability of system (1.1.4)-(1.1.6) in the case when the support of b coincide with the support of c. When the waves propagate at the same speed (i.e. a = 1), he established an exponential decay of the energy. While when the waves propagate at di erent speeds, no decay rate was discussed.
In [12], F. Alabau-Boussouira et al. considered the energy decay of the following system :
utt a u + (x; ut) + b(x)yt = 0 in R+; (1.1.7)
ytty b(x)ut = 0 in R+; (1.1.8)
u = y = 0 on R+; (1.1.9)
where a > 0 constant, b 2 C0( ; R) and (x; ut) is a non linear damping. Using an approach based on multiplier techniques, weighted nonlinear inequalities and the optimal-weight convexity method (developed in [7]), the authors established an explicit energy decay formula in terms of the behavior of the nonlinear feedback close to the origin in the case that the three following conditions are satis ed : the waves propagate at the same speed (a = 1) and the coupling coe cient b(x) is small positive (0 b(x) b0, b0 2 (0; b?] where b? is a constant depending on and on the control region) and both the coupling and the damping regions satisfying an appropriated geometric conditions named Piecewise Multipliers Geometric Conditions (PMGC, in short). But the contrary case, when the waves are not assumed to propagate with equal speed ( a is not necessarily equal to 1) and/or b is not assumed to be small and positive has been left as open question even in the linear case i:e: (x; ut) = c(x)ut. This open question will be our target in chapter 2 in the linear case.
Principal results of the chapter.
The main novelty in this chapter is that the waves are not necessarily propagating at the same speed and the coupling coe cient is not assumed to be positive and small. First, we begin to study the existence, uniqueness and regularity of the solution of our system using the semigroup approach. Let (u; ut; y; yt) be a regular solution of the system (1.1.4)-(1.1.6), its associated total energy is de ned by E(t) = 2 Z jutj2 + ajruj2 + jytj2 + jryj2 dx: (1.1.10)
Consequently, system (1.1.4)-(1.1.6) is dissipative in the sense that its energy is non-increasing with respect to the variable time t. Next, we de ne the energy space H = ( H01( ) L2( ))2 equipped, for all U = ( u; v; y; z); U = (u; v; y; z) 2 H , by the scalar product : Z (ru ru)dx + Z vvdx + Z e e e e e (U; U)H = a (ry ry)dx + Z zzdx: e e e e e
Setting U = (u; ut; y; yt), system (1.1.4)-(1.1.6) may be rewritten as :
Ut = AU; in (0; +1); U(0) = (u0; u1; y0; y1); where the unbounded operator A : D(A) H ! H is given by
D(A) = (H2( ) \ H01( )) H01( ) 2
(1.1.12)
and
AU = ( v; a u bz cv; z; y + bv ); 8 U = (u; v; y; z) 2 D(A): (1.1.13)
The operator A is m-dissipative in H and generates a C0-semigroup of contractions (etA)t 0. So, system (1.1.4)-(1.1.6) is well posed in H.
Then, we move to study the asymptotic behavior of E(t). For this aim, we assume that there exists a non empty open !c+ satisfying the following condition fx 2 : c(x) > 0g !c+ : (LH1)
On the other hand, as b(x) is not identically null and continuous, then there exists a non empty open sets !b+ [ !b such that fx 2 : b(x) > 0g !b+ and fx 2 : b(x) < 0g !b : (LH2)
We rst prove that our system is strongly stable without geometric condition. This is given by the following theorem :
Theorem 1.1.20. (Strong Stability) Assume that a > 0, condition (LH1) holds and that ! = !c+ \ !b+ 6= ; (or !c+ \ !b 6= ;). Then the semigroup of contractions (etA)t 0 is strongly stable on the energy space H, i.e. for any U0 2 H, we have lim k etAU 0kH = 0: (1.1.14) t!+1
Now, we study the energy decay rate by using a frequency domain approach combined with piecewise multiplier technique in two cases : the rst one when the waves are assumed to propagate at the same speed (i.e. a = 1) and the second case when a 6= 1.
The rst main result is the following one :
Theorem 1.1.21. (Exponential decay rate) Let a = 1. Assume that condition (LH1) holds. Assume also that the nonempty open set ! = !c+ \ !b+ (or ! = !c+ \ !b ) satis es the geometric conditions PMGC and that b; c 2 W 1;1( ). Then there exist positive constants M 1, > 0 such that for all initial data (u0; u1; y0; y1) 2 H the energy of the system
(1.1.4)-(1.1.6) satis es the following decay rate : E(t) M e tE(0); 8t > 0: (1.1.15)
Remark 1.1.22. Note that in the previous theorem we have no restriction on the upper bound and the sign of the function b. This theorem is then a generalization in the linear case of the result of [12] where the coupling coe cient considered have to satisfy 0 b(x) b 0 , b0 2 (0; b ] where b is a constant depending on and on the control region. Nevertheless, the problem still be open in the nonlinear case.
The condition of equal speed propagation is a necessary and su cient condition for the exponential stability of our system. In fact, in the case a 6= 1, we construct a sequence (Un) of elements in D(A) and a real sequence ( n) such that kUnk = 1 and k(i nI A )UnkH ! 0: Hence, the resolvent of A is not uniformly bounded on the imaginary axis. Following a result of Huang [36] and Pruss [60] we conclude that the semigroup (etA)t 0 is not uniformly stable in H. So it is natural to look for a polynomial decay of the energy. Consequently, our second main result when the wave propagate at di erent speeds (a 6= 1) can be stated as follows :
Theorem 1.1.23. (Polynomial decay rate) Let a 6= 1. Assume that condition (LH1) holds. Assume also that the nonempty open set ! = !c+ \ !b+ (or ! = !c+ \ !b ) satis es the geometric conditions PMGC and that b; c 2 W 1;1( ). Then there exists a positive constant C > 0 such that for all initial data (u0; u1; y0; y1) 2 D(A) the energy of the system (1.1.4)- (1.1.6) satis es the following polynomial decay rate :
E(t) C kU(0)kD2 (A); 8t > 0: (1.1.16)
Finally, in one space dimension (i.e. N = 1), a 6= 1 and b is a constant, we prove that there exist n0 2 N su ciently large and a sequence n of simple eigenvalues of the operator A satisfying the following asymptotic behavior
ib2 cb2 1 ; 8 jnj n0:
n = in + O (1.1.17)
2(a 1)n 2(a 1)2n2 2 n3
It follows that the obtained polynomial decay rate is optimal (see Theorem 3.4.1 in [56]). 1.1.3 Chapter 3 : Exact controllability and stabilization of lo-cally coupled wave equations The aim of this chapter is to investigate the exact controllability of the following system : utt a u + b(x)yt = c(x)v(t) in R+; (1.1.18)
ytt y b(x)ut = 0 in R+; (1.1.19)
u = y = 0 on R+; (1.1.20)
with the following initial data
u(x; 0) = u0; y(x; 0) = y0; ut(x; 0) = u1 and yt(x; 0) = y1; x 2 ; (1.1.21)
under appropriate geometric conditions. Here, a > 0 constant, b 2 C0( ; R), c 2 C0( ; R+) and v is an appropriate control. The idea is to use a result of A. Haraux in [31] for which the observability of the homogeneous system associated to (1.1.18)-(1.1.20) is equivalent to the exponential stability of system (1.1.4)-(1.1.6) in an appropriate Hilbert space. So, we provide a complete analysis for the exponential stability of system (1.1.4)-(1.1.6) in di erent Hilbert spaces.
Preceding results :
In chapter 2, we studied the stabilization of system (1.1.4)-(1.1.6) in two cases. In the rst one, when the waves are assumed propagating at the same speed (i.e. a = 1), under the assumption that the coupling region and the damping region have a non empty intersection and satisfying the PMGC condition. In this case, we established an exponential decay rate for weak initial data. On the contrary (i.e. a 6= 1 ) we rst proved the lack of the exponential stability of the system. However, under the same geometric condition, an optimal energy decay rate of type 1t was established for smooth initial data.
Our aim in this chapter is to prove the exponential stability of system (1.1.4)-(1.1.6) in two di erent Hilbert spaces by using geometric conditions more general than that used in chapter 2. And consequently, by using Proposition 2 of A. Haraux [31], we obtain the observability of the homogeneous system associated to (1.1.18)-(1.1.20).
First, we need to study the asymptotic behavior of E(t) associated to (1.1.4)-(1.1.6) and given by equation (1.1.10). For this aim, we suppose that there exists a non empty open !c+ satisfying the following condition fx 2 : c(x) > 0g !c+ : (LH1)
On the other hand, as b(x) is not identically null and continuous, then there exists a non empty open !b such that fx 2 : b(x) 6= 0g !b: (LH2)
If ! = !c+ \ !b 6= ; and condition (LH1) holds, then system (1.1.4)-(1.1.6) is strongly stable using Theorem 1.1.20, i.e. t lim k etA(u ; u ; y ; y ) kH = 0 8 (u ; u ; y ; y ) 2 H : ! + 1 0 1 0 1 0 1 0 1
Then, when the waves propagate at the same speed (i.e., a = 1), under the condition that the coupling region includes in the damping region and satisfying the called Geometric Control Condition (GCC in Short), we establish the exponential stability of system (1.1.4)-(1.1.6) given by the following theorem
Theorem 1.1.24. (Exponential decay rate) Let a = 1. Assume that conditions (LH1) and (LH2) hold. Assume also that !b !c+ satis es the geometric control conditions GCC and that b; c 2 W 1;1( ). Then there exist positive constants M 1, > 0 such that for all initial data (u0; u1; y0; y1) 2 H the energy of the system (1.1.4)-(1.1.6) satis es the following decay rate : E(t) M e tE(0); 8t > 0: (1.1.22)
Remark 1.1.25. The geometric situations covered by Theorem 1.1.24 are richer than that considered in Chapter 2 and [12]. Indeed, in the previous references, the authors consider the PMGC geometric conditions that are more restrictive than GCC. On the other hand, unlike the results in [12], we have no restriction in Theorem 1.1.24 on the upper bound and the sign of the coupling function coe cient b. This theorem is then a generalization in the linear case of the result of [12] where the coupling coe cient considered have to satisfy 0 b(x) b0, b0 2 (0; b?] where b? is a constant depending on and on the control region.
Consequently, using Proposition 2 of A. Haraux in [31], an observability inequality of the solution of the homogeneous system associated to (1.1.18)-(1.1.20) in the space (H01( ) L2( ))2 is established. This leads, by the HUM method introduced by Lions in [46], to the exact controllability of system (1.1.18)-(1.1.20) in the space (H 1( ) L2( ))2.
Furthermore, on the contrary when the waves propagate at di erent speeds, (i.e., a 6= 1), we establish the exponential stability of system (1.1.4)-(1.1.6) in the weak energy space. For this, we introduce the following weak energy space D = H01( ) L2( ) L2( ) H 1( ); equipped with the scalar product (U; U~) = Z (aru:ru~ + vv~ + yy~ + ( ) 1=2z( ) 1=2z~)dx;
for all U = (u; v; y; z) 2 D and U = (~u; v;~ y;~ z~) 2 D.
Next, we de ne the unbounded linear operator Ad : D(Ad) D ! D by
AdU = ( v; a u bz cv; z; y + bv );
D(Ad) = (H01( )\H2( )) H01( ) H01( ) L2( ) ; 8 U = (u; v; y; z) 2 D(Ad):
Its total mixed energy is de ned by
Em(t) = 12 akruk2L2( ) + kutk2L2( ) + kytk2H 1( ) + kyk2L2( ) :
Then, we move to study the asymptotic behavior of Em(t). For this aim, we need to assume that ! c+ satis es the geometric conditions PMGC, then there exist » > 0, subsets j ,
N such that if j ; :::; J ,+with Lipschitz boundary = @ and points x j 2 R i \ = ; = 1 J x j J j N j
where RN , j(xj) = j : (x x j) j (x) > 0 where j is the outward unit i 6= j and !c N [j=1 j (xj) [ n [j=1 j \ with N (O) = fx 2 R : d(x; O) < « g O f 2 g normal vector to j and that !b satis es the GCC condition and !b n [jJ=1 j : (LH3)
Now, our second main result when the waves propagate at di erent speed (i.e. a 6= 1) can be stated as follows :
Theorem 1.1.26. (Exponential decay rate) Let a 6= 1. Assume that conditions (LH1) and (LH2) hold. Assume also that !c+ satis es the geometric conditions PMGC, !b satis es GCC condition and (LH3) and b; c 2 L1( ). Then there exist positive constants M 1,> 0 such that for all initial data (u0; u1; y0; y1) 2 D the energy of system (1.1.4)-(1.1.6) satis es the following decay rate : Em(t) M e tEm(0); 8t > 0: (1.1.23)
Consequently, using Proposition 2 of A. Haraux in [31], an observability inequality of the solution of the homogeneous system associated to (1.1.18)-(1.1.20) is established. This leads, by the HUM method, to the exact controllability of system (1.1.18)-(1.1.20) in the space L2( ) H 1( ) H 1( ) (H2( ) \ H01( ))0; where the duality is according to L2( ).
Finally, we perform numerical tests in the 1-D case to insure the theoretical results obtained here and in chapter 2. In fact, the numerical results show a better behavior that the one expected by the theoretical results.
Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coe cient at interface
This chapter is devoted to study the stability of an elastic Bresse system with local Kelvin-Voigt damping and non-smooth coe cient at interface under fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The system de ned on (0; L) (0; +1) is governed by the following partial di erential equations :
8 1’tt [k1 (’x + + lw) + D1 (’xt + t + lwt)]x lk3 (wx l’) lD3 (wxt l’t) = 0;
> 2 tt [k2 x + D2 xt]x + k1 (’x + + lw) + D1 (’xt + t + lwt) = 0;
The coe cients 1; 2; k1; k2; k3 and l are positive constants. D1, D2 and D3 are positive functions over (0; L).
Preceding results :
Kelvin-Voigt material is a viscoelastic structure having properties of both elasticity and viscosity. The Kelvin-Voigt damping can be globally or locally distributed. But the case we are interested in is when it is localized on an arbitrary subinterval of the domain. The regularity and stability properties of a solution depend on the properties of the damping coe cients. Indeed, the system is more e ectivelly controled by the local Kelvin-Voigt damping when the coe cient changes more smoothly near the interface.
Table of contents :
1 Introduction
1.1 Introduction in English
1.1.1 Principal used methods
1.1.2 Local Indirect Stabilization of two coupled wave equations under geometric conditions
1.1.3 Exact controllability and stabilization of locally coupled wave equations
1.1.4 Stability of a Bresse system with local Kelvin-Voigt damping and non-smooth coecient at interface
1.2 Introduction in French
1.2.1 Methodes principales utilisees
1.2.2 Stabilite indirecte locale de deux equations d’ondes couplees sous des conditions geometriques
1.2.3 Contr^ollabilite exacte et stabilite des equations d’ondes localement couplees
1.2.4 Stabilite d’un systeme de Bresse avec amortissement local Kelvin-
Voigt et coecient non reguliere a l’interface
2 Local indirect stabilization of N-d system of two coupled wave equations under geometric conditions
2.1 Introduction
2.2 Well posedness and strong stability
2.2.1 Well posedness of the problem
2.2.2 Strong stability
2.3 Exponential stability, the case a = 1
2.4 Non uniform stability in the case a 6= 1
2.5 Polynomial stability in the case a 6= 1
2.6 Optimality of the polynomial energy decay rate
3 Exact controllability and stabilization of locally coupled wave equations
3.1 Introduction
3.1.1 Motivation and aims
3.1.2 Literature
3.1.3 Description of the chapter
3.2 Well posedeness and strong stability
3.3 Exponential stability and exact controllability in the case a = 1
3.3.1 Exponential stability
3.3.2 Observability and exact controllability
3.4 Exponential stability and exact controllability in the case a 6= 1
3.4.1 Exponential stability in the weak energy space
3.4.2 Observability and exact controllability
3.5 Numerical approximation : Validation of the theoretical results
3.5.1 Finite dierence scheme in one dimensional space
3.5.2 Numerical experiments : validation of the theoretical results
3.6 General conclusion
4 Stability of a Bresse system with local Kelvin-Voigt damping and non- smooth coecient at interface
4.1 Introduction
4.2 Well-posedness of the problem
4.3 Strong stability of the system
4.4 Analytic stability in the case of three global dampings
4.5 Exponential stability in the case of three local smooth dampings
4.6 Polynomial stability in the case of three local non smooth dampings
4.7 Lack of exponential stability
4.8 Polynomial stability in the case of one local damping .
