A DPG method for linear quadratic optimal control problems

TL;DR

This paper introduces a Discontinuous Petrov-Galerkin (DPG) method with optimal test functions for solving constrained linear quadratic optimal control problems, providing theoretical analysis and numerical validation.

Contribution

It develops a new DPG framework for constrained optimal control problems, including existence, optimality conditions, and error estimates, applicable to various PDEs.

Findings

Proved existence and uniqueness of the discrete solution.

Derived a priori and a posteriori error estimates.

Validated the method through numerical experiments.

Abstract

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented.