On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

Peter I Kogut,Rosanna Manzo,Olha P Kupenko

doi:10.1155/2020/7418707

Abstract

We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

Highlights

The main object of our study is the following optimal control problem for a nonlinear elliptic equation: Minimize Jðu, yÞ = 1 2 ð jy Ω − yd j2 dx +α ð jujp dx, pΩ ð1Þ subject to constrains−Δy = f ðyÞ + u in Ω, ð2Þ y = 0 on ∂Ω, ð3Þ u ∈ U
The novelty of this paper is that we discuss the existence of optimal pairs to optimal control problem (OCP) (1)–(4) using an indirect approach based on the classical Tikhonov regularization technique in its special implementation
Having introduced a special family of optimization problems, we show that there exists an optimal solution to the original OCP that can be attained with a prescribed level of accuracy by the sequence of optimal solutions for the regularized minimization problems

Summary

Introduction

The main object of our study is the following optimal control problem for a nonlinear elliptic equation: Minimize. The novelty of this paper is that we discuss the existence of optimal pairs to OCP (1)–(4) using an indirect approach based on the classical Tikhonov regularization technique in its special implementation. The consistency of OCP (1)–(4) and existence of optimal pairs can be established only if we impose rather strict assumptions on the original data. It was shown in [20] that the set of optimal solutions of (1)–(4) is nonempty provided. The details of the indirect approach to the study of the original optimal control problem are discussed in last section. The key points of such approach are summarized in Theorem 13

Preliminaries

CF ðzÞF

On the Tikhonov Regularization of the Original OCP

Optimality Conditions for Regularized Problem

Asymptotic Analysis of Regularized Optimal Control Problem