A Space-Time Variational Method for Optimal Control Problems

10/01/2020
by   Nina Beranek, et al.
0

We consider a space-time variational formulation of a PDE-constrained optimal control problem with box constraints on the control and a parabolic PDE with Robin boundary conditions. In this setting, the optimal control problem reduces to an optimization problem for which we derive necessary and sufficient optimality conditions. We propose to utilize a well-posed inf-sup stable framework of the PDE in appropriate Lebesgue-Bochner spaces. Next, we introduce a conforming simultaneous space-time (tensorproduct) discretization in these Lebesgue-Bochner spaces. Using finite elements in space and piecewise linear functions in time, this setting is known to be equivalent to a Crank-Nicolson time stepping scheme for parabolic problems. The optimization problem is solved by a projected gradient method. We show numerical comparisons for problems in 1d, 2d and 3d in space. It is shown that the classical semi-discrete primal-dual setting is more efficient for small problem sizes and moderate accuracy. However, the simultaneous space-time discretization shows good stability properties and even outperforms the classical approach as the dimension in space and/or the desired accuracy increases.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/06/2016

Mixed Strategy for Constrained Stochastic Optimal Control

Choosing control inputs randomly can result in a reduced expected cost i...
research
04/04/2020

Unstructured space-time finite element methods for optimal control of parabolic equations

This work presents and analyzes space-time finite element methods on ful...
research
10/21/2021

Space-time formulation and time discretization of phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization...
research
03/22/2023

A convenient inclusion of inhomogeneous boundary conditions in minimal residual methods

Inhomogeneous essential boundary conditions can be appended to a well-po...
research
05/02/2019

Lifting Vectorial Variational Problems: A Natural Formulation based on Geometric Measure Theory and Discrete Exterior Calculus

Numerous tasks in imaging and vision can be formulated as variational pr...
research
03/28/2022

Computational performance studies for space-time phase-field fracture optimal control problems

The purpose of this work are computational demonstations for a newly dev...

Please sign up or login with your details

Forgot password? Click here to reset