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arXiv · 2010.00345

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

Nina Beranek·M. Alexander Reinhold·Karsten Urban
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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.

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Nina Beranek, M. Alexander Reinhold, Karsten Urban. 2022-12-05. A Space-Time Variational Method for Optimal Control Problems: Well-posedness, stability and numerical solution. https://arxiv.org/abs/2010.00345

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