Optimal control of nonlinear systems governed by Dirichlet fractional Laplacian in the minimax framework

TL;DR

This paper addresses the optimal control of nonlinear systems involving the spectral Dirichlet fractional Laplacian, establishing conditions for optimal process existence using variational methods and the Ky Fan Theorem.

Contribution

It introduces new sufficient conditions for the existence of optimal controls in systems governed by the Dirichlet fractional Laplacian within a minimax framework.

Findings

Established existence of optimal controls under new conditions.

Applied variational methods to fractional Laplacian boundary value problems.

Utilized Ky Fan Theorem to demonstrate weak solutions.

Abstract

We consider an optimal control problem governed by a class of boundary value problem with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes is stated. The proof of the main result relies on variational structure of the problem. To show that boundary value problem with the Dirichlet fractional Laplacian has a weak solution we employ the renowned Ky Fan Theorem.