Global existence and Hadamard differentiability of hysteresis-reaction-diffusion systems

TL;DR

This paper studies the mathematical properties of reaction-diffusion systems with hysteresis, proving well-posedness, differentiability of solutions, and applying these results to an optimal control problem.

Contribution

It establishes well-posedness, Lipschitz continuity, and Hadamard differentiability of solutions for hysteresis-reaction-diffusion systems with general boundaries.

Findings

Proved well-posedness and boundedness of solutions.

Established Lipschitz continuity and Hadamard differentiability of the solution operator.

Applied results to demonstrate existence of optimal controls.

Abstract

We consider a class of semilinear parabolic evolution equations subject to a hysteresis operator and a Bochner-Lebesgue integrable source term. The underlying spatial domain is allowed to have a very general boundary. In the first part of the paper, we apply semigroup theory to prove well-posedness and boundedness of the solution operator. Rate independence in reaction-diffusion systems complicates the analysis, since the reaction term acts no longer local in time. This demands careful estimates when working with semigroup methods. In the second part, we show Lipschitz continuity and Hadamard differentiability of the solution operator. We use fixed point arguments to derive a representation for the derivative in terms of the evolution system. Finally, we apply our results to an optimal control problem in which the source term acts as a control function and show existence of an optimal solution.