椭圆型变分不等方程最优控制的必要条件

THE NECESSARY CONDITIONS OF OPTIMAL CONTROL FOR ELLIPTIC VARIATIONAL INEQUALITY

本文用 F.H.Clarke 的非光滑分析方法,在目标泛函 Lipschitz 的条件下,导出了椭圆型变分不等方程系统最优控制的必要条件,并证明有界最优控制是阀值 Q=0 的Bang-Bang 开关控制。

The optimal control system governed by an elliptic variational inequality,with the control k(x) appearing as the nonhomogeneous term,was first investigated by Avner Friedman who supposed that the index functional F(x,u)was differentiable.In this paper,the necessary condition for the same system is obtained under weaker conditions of F(x,u): (i)F(·,u)is integrable on Ω; (ii)F(x,·)satisfies the Lipschitz condition on K. Using the F.H.Clarke's nonsmooth analysis techniques,the necessary condition is expressed as —Q∈N_A(k_0) which has an obvious geometrical explanation.Even the structure of the maximizers k_0(x)is still shown to be Bang-Bang with the threshold Q=0.