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A priori error estimates for optimal control problems governed by the transient Stokes equations and subject to state constraints pointwise in time

Dmitriy Leykekhman, Boris Vexler, Jakob Wagner

TL;DR

This work develops a priori error estimates for a state-constrained, time-dependent optimal control problem governed by the transient Stokes equations, employing inf-sup stable spatial finite elements and discontinuous Galerkin time discretization. A central result bounds the fully discrete control error by $\|\bar{\mathbf q}-\bar{\mathbf q}_\sigma\|_{L^2(I\times\Omega)}$ with a rate $\mathcal{O}(\ell_k (k^{1/2}+h)/\sqrt{\alpha})$, where $\ell_k=\ln(T/k)$, and shows improved regularity for the optimal control. The paper also develops variational discretization and fully discrete schemes, proves stability and discrete optimality conditions, and provides rigorous error estimates and convergence for the Stokes state, along with comprehensive numerical experiments validating the theory. The results advance reliable discretization of transient, state-constrained PDE control problems and offer practical PDAS-based algorithms for efficient computation. $$\|\bar{\mathbf q}-\bar{\mathbf q}_\sigma\|_{L^2(I\times \Omega)} \lesssim \frac{1}{\sqrt{\alpha}} \ln\left(\frac{T}{k}\right) \left(k^{1/2}+h\right)$$ under the stated assumptions.

Abstract

In this paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an L2 functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we establish error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.

A priori error estimates for optimal control problems governed by the transient Stokes equations and subject to state constraints pointwise in time

TL;DR

This work develops a priori error estimates for a state-constrained, time-dependent optimal control problem governed by the transient Stokes equations, employing inf-sup stable spatial finite elements and discontinuous Galerkin time discretization. A central result bounds the fully discrete control error by with a rate , where , and shows improved regularity for the optimal control. The paper also develops variational discretization and fully discrete schemes, proves stability and discrete optimality conditions, and provides rigorous error estimates and convergence for the Stokes state, along with comprehensive numerical experiments validating the theory. The results advance reliable discretization of transient, state-constrained PDE control problems and offer practical PDAS-based algorithms for efficient computation. under the stated assumptions.

Abstract

In this paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an L2 functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we establish error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.
Paper Structure (15 sections, 29 theorems, 169 equations, 8 figures)

This paper contains 15 sections, 29 theorems, 169 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminary results
  3. Continuous Optimal Control Problem
  4. Finite Element Approximation of the State Equation
  5. Spatial discretization
  6. Temporal discretization: the discontinuous Galerkin method
  7. Best approximation type fully discrete error estimate for the Stokes problem in $L^\infty(I;L^2(\Omega)^d)$ norm
  8. Variational Discretization of the Optimal Control Problem
  9. Full Discretization of the Optimal Control Problem
  10. Improved regularity
  11. Numerical results
  12. Solution Algorithm
  13. Example 1
  14. Example 2
  15. Example 3

Key Result

Theorem 2.1

For ${\bf{q}} \in L^1(I;{L^2(\Omega)^d}) + L^2(I;\mathbf{V}^*)$, there exists a unique solution ${\bf{u}} \in L^2(I;\mathbf{V}) \cap C(\bar{I};\mathbf{V}_0)$ solving eq:weak_state and the following estimate holds If ${\bf{q}} \in L^2(I;{L^2(\Omega)^d})$, then ${\bf{u}} \in L^2(I;{\mathbf{V}_2}) \cap H^1(I;{\mathbf{V}_0}) \hookrightarrow C(\bar{I};\mathbf{V})$, and there holds

Figures (8)

  • Figure 1: Convergence with respect to $h$ for $k=10^{-3}$.
  • Figure 2: Convergence with respect to $k$ for different values of $h$.
  • Figure 4: Time function $\varphi(t)$.
  • Figure 5: Weight function ${\bf{w}}$.
  • Figure 6: Satisfaction of the state constraint.
  • ...and 3 more figures

Theorems & Definitions (60)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Remark 3.5
  • ...and 50 more