Analysis of optimal control problems of semilinear elliptic equations by BV-functions

TL;DR

This paper investigates optimal control problems involving semilinear elliptic equations with BV-based control costs, focusing on existence, optimality conditions, and the impact of vector norm choices on control simplicity.

Contribution

It provides new analysis on the existence and optimality conditions for BV-based controls, emphasizing the influence of vector norm selection on control structure.

Findings

BV controls promote piecewise constant solutions with few jumps

Existence and optimality conditions are established for the control problems

The choice of vector norm significantly affects control properties

Abstract

Optimal control problems for semilinear elliptic equations with control costs in the space of bounded variations are analysed. BV-based optimal controls favor piecewise constant, and hence 'simple' controls, with few jumps. Existence of optimal controls, necessary and suffcient optimality conditions of first and second order are analysed. Special attention is paid on the effect of the choice of the vector norm in the definition of the BV-seminorm for the optimal primal and adjoined variables.