Optimal control of fractional semilinear PDEs^*

¹ Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA.
² University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, College of Natural Sciences, 17 University AVE. STE 1701, San Juan, PR 00925-2537, USA.

^** Corresponding author: hantil@gmu.edu

Received: 18 December 2017
Accepted: 12 January 2019

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 ∈ (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.