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

Harbir Antil; Ciprian G. Gal; Mahamadi Warma

doi:10.3934/dcdss.2022012

Article Contents

2022, Volume 15, Issue 8: 1883-1918. Doi: 10.3934/dcdss.2022012

This issue Previous Article Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities Next Article

$ C^1 $

-VEM for some variants of the Cahn-Hilliard equation: A numerical exploration

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

1.
The Center for Mathematics and Artificial Intelligence, Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
2.
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA

Received: September 30, 2021

Published: August 2022

The first author is partially supported by NSF grants DMS-2110263, DMS-1913004 and the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036. The last author is partially supported by the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242 and the Army Research Office (ARO) under Award NO: W911NF-20-1-0115

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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 $\mathsf{first\;show}$ the existence and regularity of solutions to the forward and the associated $\mathsf{backward\;(adjoint)}$ problems. In the second part, we prove existence of optimal $\mathsf{controls }$ and characterize the associated $\mathsf{first\;order}$ 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.

Keywords:

Generalized Caputo time-fractional derivative,
time fractional sudiffusive semilinear equations,
optimal solutions,
optimality conditions.

Mathematics Subject Classification: Primary: 49J20, 49K20; Secondary: 35S15, 65R20.

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