Variational discretization of parabolic control problems on evolving surfaces with pointwise state constraints

TL;DR

This paper develops a variational discretization method for parabolic control problems on evolving surfaces, incorporating pointwise state constraints, and provides error bounds for the discretized control and state.

Contribution

It introduces a novel discretization approach for PDE-constrained control on evolving surfaces with constraints, including error analysis.

Findings

Error bounds for control approximation

Error bounds for state approximation

Effective discretization scheme for evolving surfaces

Abstract

We consider a linear-quadratic pde constrained optimal control problem on an evolving surface with pointwise state constraints. We reformulate the optimization problem on a fixed surface and approximate the reformulated problem by a discrete control problem based on a discretization of the state equation by linear finite elements in space and a discontinuous Galerkin scheme in time. We prove error bounds for control and state.