Optimal Control of Obstacle for Quasi-linear Elliptic Variational Bilateral Problems

This paper is concerned with an optimal control problem for quasi-linear elliptic variational inequality in which the bilateral obstacles are the control. The cost functional of this optimal control problem is of Lagrange type in which the pth power of Laplacian of the control appears. This feature leads to the fact that it is hard to derive the optimality system for the underlying problem. In this paper, the optimality system is established by utilizing the special structure of the approximate optimality system including the monotonicity of the leading differential operator.