Unstructured space-time finite element methods for optimal control of parabolic equations

Abstract

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, that is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residualtype error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.

An artificial neural network‐based method for the optimal control problem governed by the fractional parabolic equation

“…Recently Abdulla and Cosgrov proposed a solution for the multiphase Stefan type free boundary problem as optimal control of parabolic equation [3]. Steinbach et al used space-time finite element methods on fully unstructured simplicity space-time meshes for the numerical solution of parabolic optimal control problems [4]. Betz in Reference [5] investigated the optimal control problem governed by a nonsmooth, semilinear parabolic PDE.…”