tailieunhanh - The existence of solution and optimal control problems for the Klein-Gordon hemivariational inequality with strongly elliptic operator

In this paper, we study the optimal control problems of systems governed by Klein-Gordon hemivariational inequalities. We establish the existence of solutions and the optimal control to these problems. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2008 Vol. 53 N . 5 pp. 20-30 THE EXISTENCE OF SOLUTION AND OPTIMAL CONTROL PROBLEMS FOR THE KLEIN-GORDON HEMIVARIATIONAL INEQUALITY WITH STRONGLY ELLIPTIC OPERATOR Pham Trieu Duong and Nguyen Thanh Nam Hanoi National University of Education Abstract. In this paper we study the optimal control problems of systems governed by Klein-Gordon hemivariational inequalities. We establish the existence of solutions and the optimal control to these problems. Keywords and phrases Optimal control problem hemivariational inequality monotone operator hyperbolic equation. 1. Introduction The background of variational problems is in physics especially in solid me- chanics where non-monotone and multi-valued constitutive laws lead to hemivari- ational inequalities. In works 4 5 authors studied some applications of hemivaria- tional inequalities but there is not much literature dealing with the optimal control problems for hemivariational inequalities. The optimal control problems for operator have been established in 7 where Park and Jeong received the results on existence of solutions and optimal control. We study the optimal control problem for general strongly elliptic operator. Let Ω be a bounded domain in Rn n 2 with the boundary Ω. We first introduce the following abbreviations Q Ω 0 T Σ Ω 0 T k kk p k kW k p Ω k kp k kLp Ω . For simplicity we denote k k k kL2 Ω . We use the following symbol is scalar product in L2 Ω and will also be used for the notation of duality pairing between dual spaces. For each multi-index p p1 pn Nn p p1 . . . pn and D p u p u uxp11 .xpnn is the generalized derivative up to order p with respect to p1 x1 pxnn x x1 . xn . 20 The existence of solution and optimal control problems. Let L x t D be the following differential operator m X L x t D D p apq D q m 1 p q 1 where apq apq x t p q 1 . . . m are continuous real-valued functions on apq Q apq 1 p q aqp and lt a positive constant for all x t Q p q t 1 .