Temporally semidiscrete approximation of a Dirichlet boundary control for a fractional/normal evolution equation with a final observation

TL;DR

This paper develops and analyzes a temporally semidiscrete numerical scheme for optimal Dirichlet boundary control problems involving fractional and normal evolution equations, with rigorous convergence proofs and numerical validation.

Contribution

It introduces a novel semidiscrete approximation method that discretizes the state equation with a discontinuous Galerkin approach without discretizing the control explicitly, and proves its convergence.

Findings

Convergence of the proposed numerical scheme is rigorously established.

Numerical experiments confirm the theoretical convergence results.

The method effectively handles fractional and normal evolution equations with boundary control.

Abstract

Optimal Dirichlet boundary control for a fractional/normal evolution with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The convergence of a temporally semidiscrete approximation is rigorously established, where the control is not explicitly discretized and the state equation is discretized by a discontinuous Galerkin method in time. Numerical results are provided to verify the theoretical results.