Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

Shengyang Luan

doi:10.1155/2019/2852815

Abstract

In this paper, a state-constrained optimal control problem governed by p-Laplacian elliptic equations is studied. The feasible control set or the cost functional may be nonconvex, and the purpose is to obtain the convergence of a solution of the discretized control problem to an optimal control of the relaxed continuous problem.

Highlights

Introduction and the Optimal Control ProblemLet Ω be a bounded open convex domain of R, = 2, 3, with a Lipschitz continuous boundary Γ
Let be a compact subset of R, and we denote by U the set of measurable functions
In the eld of nite element approximations for optimal controls governed by PDEs, we refer the readers to the papers [4,5,6,7,8,9,10] and the references therein. is present paper is mainly motivated by the work of [2] where the author considered the following state equation

Summary

Introduction and the Optimal Control Problem

We rst make the following assumptions on : (S1) e function ⋅, , is measurable in Ω, ( , ⋅, ) is in 1(R), ( , ⋅, ⋅), ( , ⋅, ⋅) are continuous in R ×. Our optimal control problem can be stated as follows. In the case of no convexity assumption, optimal control problems do not have classical solutions generally, whereas the corresponding relaxed problems have solutions if some reasonable assumptions are made. Is present paper is mainly motivated by the work of [2] where the author considered the following state equation. E following result shows that problem P is stable under what cases, and which can be proved by the same arguments as that in the proof of eorem 2 [2]. It is easy to see that the continuity of h in is equivalent to the stability of P . us the lemma is proved

The Relaxed Control Problem

Numerical Approximation of the Control Problem