A Space-Time Variational Method for Optimal Control Problems: Well-posedness, stability and numerical solution

TL;DR

This paper develops a space-time variational approach for solving optimal control problems constrained by parabolic PDEs, demonstrating well-posedness, stability, and efficiency compared to traditional time-stepping methods.

Contribution

It introduces a minimal-regularity space-time variational formulation and a stable discretization method, improving computational efficiency and stability for parabolic optimal control problems.

Findings

Space-time method exhibits good stability properties.

Requires fewer degrees of freedom in time for same accuracy.

Numerical comparisons favor the proposed method over traditional time-stepping.

Abstract

We consider an optimal control problem constrained by a parabolic partial differential equation (PDE) with Robin boundary conditions. We use a well-posed space-time variational formulation in Lebesgue--Bochner spaces with minimal regularity. The abstract formulation of the optimal control problem yields the Lagrange function and Karush--Kuhn--Tucker (KKT) conditions in a natural manner. This results in space-time variational formulations of the adjoint and gradient equation in Lebesgue--Bochner spaces with minimal regularity. Necessary and sufficient optimality conditions are formulated and the optimality system is shown to be well-posed. Next, we introduce a conforming uniformly stable simultaneous space-time (tensorproduct) discretization of the optimality system in these Lebesgue--Boch\-ner spaces. Using finite elements of appropriate orders in space and time for trial and test spaces, this setting is known to be equivalent to a Crank--Nicolson time-stepping scheme for parabolic problems. Differences to existing methods are detailed. We show numerical comparisons with time-stepping methods. The space-time method shows good stability properties and requires fewer degrees of freedom in time to reach the same accuracy.