An optimal order error estimate for the variational discretization of optimal control problems in the presence of pointwise control and state constraints

TL;DR

This paper establishes an optimal order error estimate for the variational discretization of linear-quadratic optimal control problems with pointwise constraints, using a Sobolev space reformulation and the implicit multifunction theorem.

Contribution

It introduces a novel error analysis framework for variational discretization of constrained control problems employing a Sobolev space approach and the implicit multifunction theorem.

Findings

Achieves an O(h) convergence rate for finite element discretization.

Provides a bound on perturbation errors in the solutions of discretized problems.

Extends error estimates to problems with pointwise control and state constraints.

Abstract

We consider the variational discretization of a linear-quadratic optimal control problem with pointwise control and state constraints. In order to allow for a Fr\'echet smooth norm, the problem is reformulated by means of a reflexive Sobolev space instead of the space of continuous functions. The discretization of the state equation yields a family of perturbed optimal control problems, whose solutions can be computed numerically. We apply an implicit multifunction theorem (IMT) to the first order necessary conditions to proof a bound on the perturbation error for these solutions. In order to verify the abstract regularity condition of the IMT, we compute the Fr\'echet coderivative of a set-valued representation of the necessary conditions. Applying our results to an elliptic state equation in two dimensions, undergoing a simple finite element discretization, we obtain convergence of order O(h).