Optimal control of fractional semilinear PDEs

TL;DR

This paper investigates the optimal control of semilinear fractional PDEs, establishing solution boundedness, solution map Lipschitz continuity, and deriving first and second order optimality conditions under minimal regularity assumptions.

Contribution

It introduces an optimal growth condition for the nonlinearity, proving solution map Lipschitz continuity and deriving optimality conditions for control problems with fractional PDE constraints.

Findings

Solutions are bounded under minimal regularity.

Solution map is Lipschitz continuous.

First and second order optimality conditions are derived.

Abstract

In this paper we consider the optimal control of semilinear fractional PDEs with both spectral and integral fractional diffusion operators of order $2s$ with $s \in (0,1)$. We first prove the boundedness of solutions to both semilinear fractional PDEs under minimal regularity assumptions on domain and data. We next introduce an optimal growth condition on the nonlinearity to show the Lipschitz continuity of the solution map for the semilinear elliptic equations with respect to the data. We further apply our ideas to show existence of solutions to optimal control problems with semilinear fractional equations as constraints. Under the standard assumptions on the nonlinearity (twice continuously differentiable) we derive the first and second order optimality conditions.