A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

TL;DR

This paper develops a comprehensive framework for the optimal control of fractional in time semilinear PDEs, addressing existence, regularity, and optimality conditions despite challenges like the lack of semigroup property.

Contribution

It introduces a unified approach to optimal control of fractional in time PDEs, proving existence, regularity, and optimality conditions under general nonlinear assumptions.

Findings

Established existence and regularity of solutions for forward and backward problems.

Proved existence of optimal controls and derived first order optimality conditions.

Addressed the challenge of the semigroup property failure in fractional PDEs.

Abstract

We consider optimal control of fractional in time (subdiffusive, i.e., for $% 0<\gamma <1$) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we{~\textsf{first show}} the existence and regularity of solutions to the forward and the associated {\textsf{backward (adjoint)}} problems. In the second part, we prove existence of optimal {\textsf{controls% }} and characterize the associated {\textsf{first order}} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the\ semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.