具有边界流的p-Laplace方程最优控制的存在性和稳定性.doc

具有边界流的p-Laplace方程最优控制的存在性和稳定性 Existence and stability of the optimal control to p-Laplacian with convective boundary condition?? Peidong Lei??and Hang Gao School of Mathematics amp; Statistics, Northeast Normal University, Changchun 130024, People’s Republic of China Abstract. In this paper, we are concerned with the optimal control of the evolutionary p-Laplacian with the convective boundary condition taking the heat transfer coefficient as the control. We take as our cost functional the Lp-norm of the difference between the temperature attained and the desired temperature. The existence and the stability of the optimal control that minimizes the cost functional are proved. Keywords. Optimal control, convective boundary condition, nonlinear diffusion, degen- eracy. AMS Subject Classification. 35K, 93B 1 Introduction and main results In this paper, we discuss the optimal boundary control governed by the following non- linear heat conduction system?????????? yt + div ~J + λy = 0, in QT , ~J · ~n = 1∑uy, on ΓT , y(x, 0) = y0(x), in ??, (1.1) where ?? is a bounded domain of RN with boundary of class C1, QT = ?? × (0, T ), ΓT = ???? × (0, T ), ∑ ?? ???? is a fixed partial boundary with positive measure, namely, |∑| gt; 0 (| · | denotes the measure of a set), 1∑ stands for the characteristic function of the set ∑, λ ≥ 0 is a constant, ~n denotes the unit outer normal to the boundary ????, and ~J = ??|??y|p??2??y, p gt; 2 (1.2) is the heat flux. In the system (1.1), y0 is the initial temperature, y0 ∈ L∞(??) and y0 ≥ 0. The function u represents the heat transfer coefficient, which is taken as our control and acts on the system through the partial boundary ∑. ??The research was supported by the Key Project of Chinese Ministry of Education (No. 108046), and by Grant NENU-STC07007. 1 //.paper.edu Throughout this paper, the cost functional is chosen as J(u) = 1 p { β ∫∫ QT |y ?? Zd|pdxdt+ γ ∫∫ ΓT |u|pdσdt } , (1.3) where y is the corresponding temperature distribution sati