MEMORY BOUNDARY FEEDBACK STABILIZATION FOR SCHRÖDINGER EQUATIONS WITH VARIABLE COEFFICIENTS

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1 Electronic Journal of Differential Equations, Vol. 217 (217, No. 129, pp IN: URL: or MEMORY BOUNDARY FEEDBACK TABILIZATION FOR CHRÖDINGER EQUATION WITH VARIABLE COEFFICIENT ABDEELAM NAWEL, KHALED MELKEMI Communicated by Jesus Ildefonso Diaz Abstract. First we consider the boundary stabilization of chrödinger equations with constant coefficient memory feedback. This is done by using Riemannian geometry methods and the multipliers technique. Then we explore the stabilization limits of chrödinger equations whose elliptical part has a variable coefficient. We established the exponential decay of solutions using the multipliers techniques. The introduction of dissipative boundary conditions of memory type allowed us to obtain an accurate estimate on the uniform rate of decay of the energy for chrödinger equations. 1. Introduction Let be an open bounded domain in R n with boundary Γ :=. It is assumed that Γ consists of two parts Γ and such that Γ,, Γ =. We consider the mixed problem for chrödinger equation where y t iay = in R +, (1.1 y(, x = y (x in, (1.2 y = on Γ R +, (1.3 = u on R +, (1.4 Ay = ( a ij (x, x j the functions a ij = a ji are C functions in R n, = a ij (x, t ν i, 21 Mathematics ubject Classification. 93D15, 35J1, 35B4, 53C17. Key words and phrases. chrödinger equation; exponential stabilization; boundary condition of memory type; Riemannian geometry. c 217 Texas tate University. ubmitted February 2, 217. Published May 11,

2 2 A. NAWEL, K. MELKEMI EJDE-217/129 ν = (ν 1, ν 2,..., ν n is the unit normal of Γ pointing toward the exterior of, ν A = Aν, and A = (a ij is a matrix function. We assume that a ij (xξ i ξ j > α 1 for some positive constant α 1, and u = t n ξ 2 i x, ζ C n, ζ, (1.5 k(t sy s (sds by t. Then we transform the boundary condition in. uppose that y = on, by integration by parts t k(t sy t (sds = [k(t sy(s t + = = t t Throughout the paper we assume that u = t t k (t sy(sds k (t sy(sds + k(y(t k(ty k (t sy(sds + k(y(t. k (t sy(sds k(oy(t by t. (1.6 where k : R + R + C 2 (R +, L ( and b : R + L (. We define the corresponding energy functional by t 2E(t := g y 2 g d + k y 2 d k (t s y(t y(s d ds. (1.7 Our goal is to stabilize the system (1.1 (1.4 and (1.6; to find a suitable feedback u = F (x, y t such that the energy (1.7 decays to zero exponentially as t + for every solution y of which E( < +. The approach adopted uses Riemannian geometry. This method was first introduced to boundary-control problems by Yao [1 for the exactly controllability of wave equations. The stabilization of partial differential equations has been considered by many authors. The asymptotic behaviour of the wave equation with memory and linear feedbacks with constant coefficients has been studied by Guesmia [4, and by Aassila et al [1 in the nonlinear case. This study has been generalised by Chai & Guo [2 for variable coefficients by using a very different method, namely, the Riemannian geometry method. On the other hand, the stabilization of the chrödinger equations has been studied by Machtyngier & Zuazua [9 in the Neumann boundary conditions, and by Cipolatti et al [3 with nonlinear feedbacks. This study has been considered by Lasiecka & Triggani [8 with constant coefficients acting in the Dirichlet boundary conditions. The objective of this work, we consider the boundary stabilization for system (1.1 (1.4 and (1.6 with variable coefficients and memory feedbacks by using multipliers techniques.

3 EJDE-217/129 MEMORY BOUNDARY FEEDBACK TABILIZATION 3 Our paper is organized as follows. In subsection 1.1, we introduce some notation and results on Riemannian geometry. Our main results are studied in section 2. ection 3 is devoted to the proof of the main results Notation. The definitions here are standard and classical in the literature, see Hebey [5. Let A(x and G(x, respectively, be the coefficient matrix and its inverse A(x = (a ij (x, G(x = [A(x 1 = (g ij (x (1.8 for i, j = 1,..., n; x R n. Euclidean metric on R n. Let (x 1,..., x n be the natural coordinate system in R n. For each x R n, denote X Y = α i β i, X 2 = X X, for X = α i, Y = β i T x R n. For f C 1 ( and X = n α i, we denote by f f = and div (X = the gradient of f and the divergence of X in the Euclidean metric. α i (x (1.9 Riemannian metric. For each x R n, define the inner product and the corresponding norm on the tangent space T x R n by g(x, Y = X, Y g = X G(xY = g ij (xα i β j X 2 g = X, Y g for X = α i, Y = β i T x R n Then (R n, g is a Riemannian manifold with a Riemannian metric g. Denote the Levi-Cevita connection in metric g by D. Let H be a vector field on (R n, g. The covariant differential DH of H determines a bilinear form on T x R n T x R n. For each x R n, by DH(X, Y = D X H, Y g, X, Y T x R n where D X H is the covariante derivative of H with respect to X. The following lemma provides some useful equalities. Lemma 1.1 ([1, lemma 2.1. Let f, h C 1 ( and let H, X be a vector field on R n. Then using the above notation, we have (i H(x, A(xX(x g = H(xX(x, x R n (1.1 (ii The gradient g f of f in the Riemannian metric g is given by ( n g f(x = a ij (x f = A(x f. (1.11 x j (iii j=1 = (A(x y.ν = g y.ν. (1.12

4 4 A. NAWEL, K. MELKEMI EJDE-217/129 (iv g f, g H g = g f(h = f A(x h. (1.13 (v g f, g H(f = DH( g f, g f div ( g f 2 gh(x 1 2 gf 2 g div (H x R n. (1.14 (vi Ay = (a ij (x x j = div (A(x y = div ( g y, y C 2 (. ( tatement of main result To obtain the boundary stabilization of problem (1.1 (1.4 and (1.6, we assumed that there exists a vector field H on the Riemannian manifold (R n, g such that: X T x R n, a >, D X H, X g a X 2 g, (2.1 k and k, on Γ R +. Moreover, ϕ = H(x ν < on Γ, (2.2 H(x ν on. (2.3 inf ( k, (2.4 (x,t Γ R + k. (2.5 We have the following result of existence and uniqueness of weak solution to (1.1 (1.4 and (1.6. Theorem 2.1. For all initial data y V = H ( = {y H 1 (; y = on Γ }, problem (1.1 (1.4 and (1.6 admits a unique global weak solution y C(R +, V. Furthermore, if y H 3 ( H ( and = 1 2 ky( on, then the solution has the regularity y C 1 (R +, V. Proof. Existence of a solution is proved using Galerkin method [6. uppose y is the unique global weak solution of problem (1.1 (1.4 and (1.6. The variational formulation of the problem is y t v d = i Ayv d = i = i = i ( a ij (x v d x j a ij (x ν i v d i v d i a ij (x v xj d a ij (x v xj d

5 EJDE-217/129 MEMORY BOUNDARY FEEDBACK TABILIZATION 5 for v V, and y t v d = i ( i t k (t sy(sds k(y(t by t v d a ij (x v xj d. We introduce the following notation:, is the scalar product in L 2 (, a(y, v = a ij (x v xj d, t β(t, y, v = ( k (t sy(sds k(y(tv d β(, y, v =. We have y t, v + ia(y, v + iβ(t, y, v = i by t, v Uniqueness. Let y and z be two solutions. then w = y z satisfies w = t w t iaw = in, + (2.6 w(, x = in (2.7 w = on, + Γ (2.8 k (t sw(sds k(w(t bw t, on, + [. (2.9 Multiplying (2.6 by w t, integrating over. Applying the Green formula, we obtain w t 2 d i a ij (x w ν i w t dγ + i a ij (x w w t xj d =. Taking the imaginary part of above equation, Re w t dγ ν 1 = Re g w g w t d A = 1 [ d 2 dt gw 2 g d = 1 d g w(t 2 g d. 2 dt For t =, we have t g w 2 g d + k w 2 d k (t s w(t w(s 2 d ds = so t g w 2 g d = k w 2 d + k (t s w(t w(s 2 d ds. We deduce g w 2 g d and so g w =, this with the condition at limit, w =

6 6 A. NAWEL, K. MELKEMI EJDE-217/ Existence of solutions. Let (e n n ℵ be a set of functions in V that form an orthonormal basis for L 2 (. Let V m be the space generated by (e 1,..., e m, and y m (t = be a solution of the Cauchy problem With v = y m t, we have m α im (te i. y m t, v + ia(y m, v + iβ(t, y m, v = i by m t, v. y m t, y m t + ia(y m, y m t + iβ(t, y m, y m t = i by m t, y m t. Taking the imaginaire part, it results ( n Re a ij (x m m t xj d t ( = b yt m 2 d Re k (t sy m (syt m (tds d ( t Re k(y m (tyt m (tds d. However, ( b yt m 2 d Re t k (t sy m (sy m t (tds d ( t Re k(y m (tyt m (t d = b yt m 2 d + 1 k y m 2 d 2 1 t k (t s y m (t y m (s 2 ds d d 2 dt[ k y m 2 1 d [ t d + k (t s y m (t y m (s 2 ds d, 2 dt so ( n Re a ij (x m m t xj d = b yt m 2 d 1 k y m 2 d 1 t k (t s y m (t y m (s 2 ds d d [ k y m 2 d + 1 d [ t k (t s y m (t y m (s 2 ds d. 2 dt 2 dt We have b yt m 2 d + 1 k y m 2 d t k (t s y m (t y m (s 2 ds d <,

7 EJDE-217/129 MEMORY BOUNDARY FEEDBACK TABILIZATION 7 thus ( n Re a ij (x m m t xj d 1 d [ k y m 2 d + 1 d [ 2 dt 2 dt We deduce that On the other hand, Therefore, ( n Re from where ( n Re t k (t s y m (t y m (s 2 ds d. a ij (x m m t xj d. a ij (x m m t xj d = 1 d ( n a ij (x m m 2 dt xj d = 1 d ( g y m 2 g d. 2 dt 1 g y m (T 2 g d 1 g y m ( 2 g d, 2 2 y m 2 V y m 2 V and v m v in V, or yt m, v m + ia(y m, v m + iβ(t, y m, v m = i byt m v m d. (2.1 For ζ D(, T [, we put so we have ψ m = ζv m and ψ = ζv ; ψ m ψ in L 2 (, T, V. Multiplying (2.1 by ζ and integrating over, T [, we find T = i T yt m, ψ m dt + i T by m t ψ m d dt. T a(y m, ψ m dt + i β(t, y m, ψ m dt Passing to the limit, T T T T y, ψ t dt + i a(y, ψdt + i β(t, y, ψdt = i byψ t d dt for all v V and all ζ D(, T [. y C, T [, V.

8 8 A. NAWEL, K. MELKEMI EJDE-217/129 Regularity. We have ytt m, v + i + i ( t a ij (x m t k (t sy m (sds k(yt m (t v d v xj d i bytt m v d =. Putting v = ytt m, t ytt m, ytt m + i ( k (t sy m (sds k(yt m (tytt m d + i a ij (x m t m tt xj d i bytt m ytt m d =. Taking the imaginary part, ( n Re a ij (x m t m tt xj d t ( = b ytt m 2 d Re k (t syt m (sytt m (tds d ( t Re k(yt m (tytt m (tds d, we have t b ytt m 2 d Re( k (t sy m (sytt m (tds d t Re( k(yt m (tytt m (t d = b ytt m 2 d 1 k yt m 2 d t k (t s yt m (t + y m (s 2 ds d 2 1 d [ k yt m 2 d 1 t k (t s yt m (t + y m (s 2 ds d. 2 dt 2 We deduce that b ytt m 2 d 1 k yt m 2 d so Re( a ij (x m t tt m xj d 1 d 2 dt ( k yt m 2 d t t k (t s y m t (t+y m (s 2 ds d ; k (t s y m t (t + y m (s 2 ds d.

9 EJDE-217/129 MEMORY BOUNDARY FEEDBACK TABILIZATION 9 On the other hand, ( n Re and 1 d 2 dt ( 1 d 2 dt g y m t 2 g d a ij (x m t m tt xj d = 1 2 ( k yt m 2 d Therefore, 1 d [ k yt m 2 d dt 2 1 d ( 2 dt We have t t d ( dt g yt m 2 g d k (t s y m t (t + y m (s 2 ds d. k (t s y m t (t + y m (s 2 ds d, g yt m 2 g d. 1 g yt m (T 2 g d 1 g yt m ( 2 g d, 2 2 y m 2 V y m 2 V Our main result is the following theorem. Theorem 2.2. Assume (1.5 (1.7 and (2.1 (2.5, and that the problem y t iay = on y = in Σ = in Σ has y = as the unique solution. Then for all given initial data y H 1 Γ (, there exist two positive constants M and ω such that E(t Me ωt E(, for t >. 3. Proof of main result For simplicity, we assume that y is a weak solution. argument, Theorem 2.2 still holds for a weak solution. By a classical density Lemma 3.1. The energy defined by (1.7 is decreasing and satisfies T E(T E( = y t 2 d dt + 1 T k y 2 d dt 2 1 T t k (t s y(t y(s 2 d ds dt, 2 whenever < T <. Proof. Differentiating the energy E( defined by (1.7 and using Green s second theorem, we have 2E (t = g y t g y d + g y g y t d + k y 2 d

10 1 A. NAWEL, K. MELKEMI EJDE-217/129 Then Then t + 2 Re ky t y d k (t s y(t y(s 2 d ds t k (t s d dt (y(t y(s(y(t y(sds d. E (t = Re g y t, g y g d + 1 k y 2 d Re ky t y d 1 t k (t s y(t y(s 2 d ds 2 t Re k (t sy t (t(y(t y(s d ds. E (t = Re y t d + 1 k y 2 d 2 + Re ky t y d 1 t k (t s y(t y(s 2 d ds 2 t Re k (t sy t (t(y(t y(s d ds By the boundary condition, we have Re yy t (k k( t k (t sds d =. Then we find that E (t = b y t 2 d + 1 k y 2 d 2 1 t k (t s y(t y(s 2 d ds. 2 This completes the proof. Lemma 3.2. For all < T < we have T ( 2 H ν T ν A (x 2 d ds g y 2 ghν d ds g T ( T + 2 Re H(y dγ1 ds + Im yy t Hυ d ds T T = 2 Re DH( g y, g y d ds g y 2 g div H d ds + Im y t H(y d T T + Im y t y div H d ds Proof. We multiply (1.1 by H y and integrate over, T [, to obtain T T y t H y d ds i AyH y d ds =. (3.1

11 EJDE-217/129 MEMORY BOUNDARY FEEDBACK TABILIZATION 11 We have T = = y t H y d ds yh y d T T yh y d T T Γ yh y t d ds ubstituting (3.2 in (3.1, we obtain yh y d T T yy t H ν dγds Hence [ T i (AyH y + AyH y d ds T 2 Re = Im + Im T Γ T yy t H ν dγds + y t div(y.h d ds. AyH y d ds yh y d T T Im y t y div H d ds T + y t y div H d ds =. Γ yy t H ν dγ ds Using (1.14, we rewrite the integral on the left-hand side of (3.3 as T AyH y d ds T = Γ T T T H(y dγds DH( g y, g y d ds div( g y 2.H d ds g y 2 g div H d ds Recalling the boundary condition (1.2, on Γ we have (3.2 (3.3 (3.4 y = y t =, g y 2 1 g = ν A (x 2 ( 2, H(y = H ν (. (3.5 g ν A (x 2 g Thus using this equation and (1.3, we find that this simplifies the sought-after identity. Completion of the proof of theorem 2.1. et C = ( H α 1 sup 2 2, x H ν C1 = α 1 2 sup x div H, C 2 = α 1 2ε sup x g (div H sup div H + εα 1 x [ k( L (Γ C 3 = h L eδf ( 2 sup H(x, x

12 12 A. NAWEL, K. MELKEMI EJDE-217/129 where h(x = that T 2a T k( δ(1+ek (, x. From assumptions (2.1, (2.2 (2.3, we deduce + Im T Im g y 2 g d ds T g y 2 gh ν d ds + 2 Re T yy t H ν d ds + g y 2 g div H d ds yh(y d T T Im y t y div HdQ ( H(y dγ1 ds Also for an arbitrary positive constant ε we have the estimates (3.6 T T g y 2 gh ν d ds + 2 Re T ( 2 C dγ1 ds, T Im yy t H ν d ds Im sup x H T y 2 d ds 2α 1 T + 2α 1 sup H ( εα1 + x 2 sup x T g y 2 g div H d ds Im T C 1 ( α1 + 2ε sup x ( 2 dσ 1 3[ Σ 1 ( H(y dγ1 ds yh(y d T y t 2 d ds + ε sup H(x E( x H(x y 2 C([,T,L 2 (, ( 2 dγ1 ds + ε 2 T T y t y div H d ds g y 2 g d ds g (div H + 1 T 4 sup div H y 2 d ds, x T ( t 2 k (t sy(sds dγ1 ds + T k( y 2 d ds + T y t 2 d ds. (3.7 (3.8 (3.9 (3.1 Now using a compactness-uniqueness argument in the style of [7, we deduce T y 2 C([,T,L 2 ( y t 2 d ds (3.11

13 EJDE-217/129 MEMORY BOUNDARY FEEDBACK TABILIZATION 13 Inserting (3.7, (3.8, (3.9 and (3.11 in (3.6, leads to T 2a g y 2 g d ds [ T 3(C + C 1 ( [ T + 3(C + C 1 t k (t sy(sds 2 d ds T k( y 2 d ds + by t 2 d ds T + C 2 y 2 d ds + ε T g y 2 g d ds 2 T + ε sup H(x E( + 2α 1 sup H y t 2 d ds x x + ( εα 1 2 sup H(x y 2 C([,T,L 2 ( x (3.12 Choosing ε so that 4a ε >, we obtain where T (4a ε E(tdt [3C 3 (C + C 1 + ε sup H(x E( + 2α 1 sup H x x + [3(C + C 1 k( 2 L + C 2 1 ϕ E( + [3(C + C 1 + εα 1 2 sup H(x sup b E(, x x T E(tdt CE(, 1 sup x Γ1 b E( 1 C = (4a ε [3C 1 3(C + C 1 + ε sup H(x + 2α 1 sup H x x sup x Γ1 b + [3(C + C 1 + εα 1 2 sup H(x sup b x x + [3(C + C 1 k( 2 L + C 2 1 ϕ. Letting T +, we obtain for every R +, + E(tdt CE(. The desired conclusion follows now from Komornik [6, Theorem 8.1. Acknowledgments. I would like to thank to Prof.. E. Rebiai for support received about the regularity questions for the chrodinger equation.

14 14 A. NAWEL, K. MELKEMI EJDE-217/129 References [1 M. Aassila, M. M. Cavalcanti, J. A. oriano; Asymptoic stability and energy deay rate for solution of wave equation with memory in a star shaped domain, IAM J. Control Optim., 38 (2, [2. Chai, Y. Guo; Boundary stabilization of wave equations with variable coefficients and memory, Differential and Integral Equations, 17 (24, [3 R. Cipolotti, E. Machtyngier, an Perdo iqueira; Nonlinear boundary feedback stabilization for chrödinger equations, Differrential and Integral Equations, 9 (1996, [4 A. Guesmia; Thèse de doctorat (Chapitre 8, IRMA, trasbourg, (2. [5 E. Hebey; Introduction à l analyse non liné aire sur les variétés, Editeur Diderot, Paris, [6 V. Komornik; Exact controlability and stabilization. The multiplier method, Masson, Paris, [7 I. Lasiecka, R. Triggiani; Optimal reglarity, exact controllability and uniform stabilization of chrödinger equations with Dirichlet control, Differrential and Integral Equations, 5 (1992, [8 J. L. Lions; Quelques méthodes de résolution des problèmes aux limites non linèaires, Dunod, Paris, [9 E. Machtyngier, E. Zuazua; tabilization of chrödinger equation, Portugaliae Mathematica, 51 (1994, [1 P. F. Yao; Observabiliy inequality for exact controllability of wave equations with variable coefficients, IAM J. Control Optim, 37 (1999, Abdesselam Nawel Department of Mathematics, University of Laghouat, Algeria address: nawelabedess@gmail.com Khaled Melkemi Department of Mathematics, University of Batna II, Algeria address: k.melkemi@univ-batna2.dz