Exact Neumann boundary controllability for problems of transmission of the wave equation

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 ˆ 0 in Q; y…x; 0† ˆ y0…x†; y0…x; 0† ˆ y1…x† in ; @y2 @ ˆ g on ; y1 ˆ y2; a1 ˆ @y1 @ ˆ a2 @y2 @ on 1; 8><>>: …1:1† satis®es y…x;T; g† ˆ y0…x;T; g† ˆ 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† ˆ 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 ˆ f in Q; u…x; 0† ˆ u0…x†; u0…x; 0† ˆ u1…x† in ; @u2 @ ˆ 0 on ; u1 ˆ u2; a1 ˆ @u1 @ ˆ a2 @u2 @ on 1; 8><>>: …2:1† where u1=uj 1 and u2=uj 2. Set H2… 1; 2† ˆ fu : u 2 H1… †; ui ˆ uj i 2 H2… i†; i ˆ 1; 2; a1 @u1 @ ˆ a2 @u2 @ on ÿ1; @u2 @ ˆ 0 on ÿg …2:2† with the norm k u kH2… 1; 2†ˆ ‰ k u kH1… † ‡ k u1 kL2… 1† ‡ k u2 kL2… 2†Š: …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 Š;H1… †† \ C1…‰0;T Š;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… †Š: …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 Š;H2… 1; 2†† \ C1…‰0;T Š;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… ††Š: …2:7† 3. Basic inequalities. We adopt the notation used in [6,7] as follows. Let x02Rn, and set m…x† ˆ xÿ x 0 ˆ …xk ÿ xk†: ÿ…x 0† ˆ fx 2 ÿ : m…x† …x† ˆ mk…x† k…x† > 0g ÿ …x 0† ˆ ÿÿ ÿ…x 0† ˆ fx 2 ÿ : m…x† …x† 0g …x 0† ˆ ÿ…x 0† …0;T † …x 0† ˆ ÿ …x 0† …0;T † R…x 0† ˆ max x2 jm…x†j ˆ maxx2 j kˆ1…xk ÿ xk†j 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† ˆ 1 2 … ‰ ju 0…x; t†j2 ‡ A…x†jruj2 Š 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 ˆ 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…x†j 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 j2†d ÿ … Q qk @u @xk fdxdt; …3:1† where u 0…t†; qk @u…t† @xk ˆ … u 0…t†qk @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 ˆ u 0…t†; qk @u…t† @x j0 ‡ … Q A…x†j @u @x j2 @q @x dxdt ‡ 1 2 … Q @q @x j u 0 j2 ÿ A…x†j @u @x j2 dxdtÿ a1 1ÿ a1 a2 … 1 q j @u1 @ j2d ÿ 1 2 … 1 q …a2j ru2 j2 ÿ a1j ru1 j2†d 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 ˆ … Q qk @u @xk fdxdt: …3:2† Integrating by parts, we obtain 128 WEIJIU LIU and GRAHAM H. WILLIAMS