Optimal potentials for problems with changing sing data

TL;DR

This paper proves the existence of optimal potentials in elliptic PDE control problems without sign restrictions, extending previous results, and demonstrates the findings with numerical simulations using FreeFem++.

Contribution

It removes the sign assumption on data for existence of optimal potentials in Schrödinger-type PDE control problems, broadening applicability.

Findings

Existence of optimal potentials established without sign restrictions.

Numerical simulations validate theoretical results.

Extended applicability of control solutions to more general data conditions.

Abstract

We consider optimal control problems where the state equation is an elliptic PDE of a Schr\"odinger type, governed by the Laplace operator $-\Delta$ with the addition of a potential V, and the control is the potential V itself, that may vary in a suitable admissible class. In a previous paper (Ref. [7]) an existence result was established under a monotonicity assumption on the cost functional, which occurs if the data do not change sign. In the present paper this sign assumption is removed and the existence of an optimal potential is still valid. Several numerical simulations, made by FreeFem++, are shown