Exact Neumann boundary controllability for problems of transmission of the wave equation
نویسندگان
چکیده
Using the Hilbert Uniqueness Method, we study the problem of exact controllability in Neumann boundary conditions for problems of transmission of the wave equation. We prove that this system is exactly controllable for all initial states in L( ) (H( ))0. 1. Introduction. Throughout this paper, let be a bounded domain (open, connected, and nonempty) in R(n 1) with a boundary ÿ=@ of class C, and 1 given with 1 and ÿ1=@ 1 of class C. Let T>0. Set 2= ÿ 1, Q= (0,T ), Q1= 1 (0,T ), Q2= 2 (0,T ), =ÿ (0,T ), 1=ÿ1 (0,T ). In [6], Lions studied the problem of exact controllability with Dirichlet boundary conditions for problems of transmission of the wave equation by introducing the Hilbert Uniqueness Method (HUM for short). Later, Nicaise [10±12] further considered this problem in R with singularities. In this paper, we consider the following Neumann boundary controllability problem in R: For suitable times T>0 and every initial condition {y , y }, does there exist a control function g such that the solution y=y(x,t;g) of the Neumann boundary value problem y00 ÿ A x y 0 in Q; y x; 0 y0 x; y0 x; 0 y1 x in ; @y2 @ g on ; y1 y2; a1 @y1 @ a2 @y2 @ on 1; 8><>>: 1:1 satis®es y x;T; g y0 x;T; g 0 in ? 1:2 In (1.1), y1=yj 1, y2=yj 2, is the unit normal of ÿ or ÿ1 pointing towards the exterior of or 1, and A(x) is given by A x a1; x 2 1; a2; x 2 2; where a1, and a2 are positive constants. We will prove that if 1 is star-shaped and a2 a1, then for all initial states fy0; y1g 2 L2 H1 0; Glasgow Math. J. 41 (1999) 125±139. # Glasgow Mathematical Journal Trust 1999. Printed in the United Kingdom there exists a control function g such that the solution y=y (x,t;g) of (1.1) satis®es (1.2). Here and in the sequel, H( ) always denotes the usual Sobolev space for s2R. The plan for the rest of this paper is as follows. In Section 2, we present the theorem about the existence and uniqueness of solutions of the problem of transmission. The estimates for the solutions (i.e., the so-called ``inverse inequality'') are given in Section 3. The main theorems of this paper are established in Section 4. 2. Homogeneous boundary problems. Consider the following homogeneous boundary problem u00 ÿ A x u f in Q; u x; 0 u0 x; u0 x; 0 u1 x in ; @u2 @ 0 on ; u1 u2; a1 @u1 @ a2 @u2 @ on 1; 8><>>: 2:1 where u1=uj 1 and u2=uj 2. Set H2 1; 2 fu : u 2 H1 ; ui uj i 2 H2 i; i 1; 2; a1 @u1 @ a2 @u2 @ on ÿ1; @u2 @ 0 on ÿg 2:2 with the norm k u kH2 1; 2 k u kH1 k u1 kL2 1 k u2 kL2 2: 2:3 The well-posedness of (2.1) is by now well known ([3], Vol.5, Chap. XVIII] and [4]). We have the following result. Theorem 2.1. (i) Suppose ÿ and ÿ1 are Lipschitz. Then, for any initial condition (u0,u1)2H1( ) L( ) and f2L1(0,T;L2( )), problem (2.1) has a unique weak solution u with u 2 C 0;T ;H1 \ C1 0;T ;L2 : 2:4 Moreover, there exists a constant C>0 such that for every t2[0,T ] k u t kH1 k u0 t kL2 C k u kH1 k u kL2 k f kL1 0;T;L2 : 2:5 (ii) Suppose ÿ and ÿ1 are of class C . Then for any initial condition (u0,u1)2H2( 1, 2) H( ) and f2L1(0,T;H1( )), problem (2.1) has a unique strong solution u with 126 WEIJIU LIU and GRAHAM H. WILLIAMS u 2 C 0;T ;H2 1; 2 \ C1 0;T ;H1 : 2:6 Moreover, there exists a constant C>0 such that for every t2[0,T ] k u 0 t kH1 k u t kH2 1; 2 C k u kH1 k u kH2 1; 2 k f kL1 0;T;H1 : 2:7 3. Basic inequalities. We adopt the notation used in [6,7] as follows. Let x02Rn, and set m x xÿ x 0 xk ÿ xk: ÿ x 0 fx 2 ÿ : m x x mk x k x > 0g ÿ x 0 ÿÿ ÿ x 0 fx 2 ÿ : m x x 0g x 0 ÿ x 0 0;T x 0 ÿ x 0 0;T R x 0 max x2 jm xj maxx2 j k1 xk ÿ xkj 1 2: where denotes the outward unit normal to ÿ. We de®ne the energy of the solution u of (2.1) by E t 1 2 ju 0 x; tj2 A xjruj2 dx; If f=0, then we have the classical result (see [6,9]) E t E 0: The following identities are essential for establishing the follow-up inverse inequalities. Lemma 3.1. Let q=(qk) a vector ®eld in [C( )]. Suppose u is the strong solution of (2.1) in the sense of (ii) of Theorem 2.1. Then the following identity holds: EXACT NEUMANN BOUNDARY CONTROLLABILITY 127 1 2 qk k ju 2j ÿ a2jr u2j ÿ d u 0 t; qk @u t @xk j0 Q A x @u @xj @qk @xi @u @xk dxdt 1 2 Q @qk @xk ju 0j2 ÿ A xj ru j2 ÿ dxdt ÿ a1 1ÿ a1 a2 1 qk kj @u1 @ j2d ÿ 1 2 1 qk k a2j ru2 j2 ÿ a1j ru1 j2d ÿ Q qk @u @xk fdxdt; 3:1 where u 0 t; qk @u t @xk u 0 tqk @u t @xk dx; and rs u={sju}j=1 n denotes the tangential gradient of u on ÿ. (See [6, p.137].) Remark 3.2. If n=1, then (3.1) becomes 1 2 q j u 2 j2d u 0 t; qk @u t @x j0 Q A xj @u @x j2 @q @x dxdt 1 2 Q @q @x j u 0 j2 ÿ A xj @u @x j2 dxdtÿ a1 1ÿ a1 a2 1 q j @u1 @ j2d ÿ 1 2 1 q a2j ru2 j2 ÿ a1j ru1 j2d ÿ Q q @u @x fdxdt: 3:10 This is a generalisation of the identity in Remark 1.5 of [6]. Proof. Multiplying (2.1) by qk @u @xk and integrating on Q, we have Q qk @u @xk u 00dxdtÿ Q qk @u @xk A x udxdt Q qk @u @xk fdxdt: 3:2 Integrating by parts, we obtain 128 WEIJIU LIU and GRAHAM H. WILLIAMS
منابع مشابه
Exact Controllability for a Wave Equation with Mixed Boundary Conditions in a Non-cylindrical Domain
In this article we study the exact controllability of a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. The fixed endpoint has a Dirichlet-type boundary condition, while the moving end has a Neumann-type condition. When the speed of the moving endpoint is less than the characteristic speed, the exact controllability of this equation is established by Hi...
متن کاملLocal exact controllability of the 2D-Schrödinger-Poisson system
In this article, we investigate the exact controllability of the 2DSchrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of R with a Poisson equation for the electrical potential. The control acts on the system through a Neumann boundary condition on the potential, locally distributed on the boundary of the space domain. We prove several results, with or without n...
متن کاملA Boundary Meshless Method for Neumann Problem
Boundary integral equations (BIE) are reformulations of boundary value problems for partial differential equations. There is a plethora of research on numerical methods for all types of these equations such as solving by discretization which includes numerical integration. In this paper, the Neumann problem is reformulated to a BIE, and then moving least squares as a meshless method is describe...
متن کاملBoundary controllability for the quasilinear wave equation
We study the boundary exact controllability for the quasilinear wave equation in the higher-dimensional case. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way tha...
متن کاملBoundary controllability of a coupled wave/Kirchoff system
We consider two problems in boundary controllability of coupled wave/Kirchoff systems. Let Ω be a bounded region in R, n ≥ 2, with Lipschitz continuous boundary Γ. In the motivating structural acoustics application, Ω represents an acoustic cavity. Let Γ0 be a flat subset of Γ which represents a flexible wall of the cavity. Let z denote the acoustic velocity potential, which satisfies a wave eq...
متن کاملA Collocation Method with Modified Equilibrium on Line Method for Imposition of Neumann and Robin Boundary Conditions in Acoustics (TECHNICAL NOTE)
A collocation method with the modified equilibrium on line method (ELM) forimposition of Neumann and Robin boundary conditions is presented for solving the two-dimensionalacoustical problems. In the modified ELM, the governing equations are integrated over the lines onthe Neumann (Robin) boundary instead of the Neumann (Robin) boundary condition equations. Inother words, integration domains are...
متن کامل