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Analysis and approximation to parabolic optimal control problems with measure-valued controls in time

Wei Gong, Dongdong Liang

TL;DR

This work studies a parabolic optimal control problem with time-measure controls, formulating a reduced cost and a state equation driven by a measure-valued control. It develops a space-time finite element discretization with a Petrov–Galerkin time scheme and variational discretization for the control, and proves well-posedness and first-order optimality conditions that reveal a time-sparsity structure. The paper provides comprehensive error analysis, showing a priori estimates and weak-* convergence of the control, with state-convergence rates of $O(h^{1/2}+\tau^{1/4})$ and cost-convergence $O(h+\tau^{1/2})$, and demonstrates that the discrete problem has a unique solution in the discretized control space. These results establish rigorous convergence and regularity for numerical approximations of measure-valued, sparsity-promoting parabolic controls, enabling accurate impulse-like actuation in evolving systems.

Abstract

In this paper, we investigate an optimal control problem governed by parabolic equations with measure-valued controls over time. We establish the well-posedness of the optimal control problem and derive the first-order optimality condition using Clarke's subgradients, revealing a sparsity structure in time for the optimal control. Consequently, these optimal control problems represent a generalization of impulse control for evolution equations. To discretize the optimal control problem, we employ the space-time finite element method. Here, the state equation is approximated using piecewise linear and continuous finite elements in space, alongside a Petrov-Galerkin method utilizing piecewise constant trial functions and piecewise linear and continuous test functions in time. The control variable is discretized using the variational discretization concept. For error estimation, we initially derive a priori error estimates and stabilities for the finite element discretizations of the state and adjoint equations. Subsequently, we establish weak-* convergence for the control under the norm $\mathcal{M}(\bar I_c;L^2(ω))$, with a convergence order of $O(h^\frac{1}{2}+τ^\frac{1}{4})$ for the state.

Analysis and approximation to parabolic optimal control problems with measure-valued controls in time

TL;DR

This work studies a parabolic optimal control problem with time-measure controls, formulating a reduced cost and a state equation driven by a measure-valued control. It develops a space-time finite element discretization with a Petrov–Galerkin time scheme and variational discretization for the control, and proves well-posedness and first-order optimality conditions that reveal a time-sparsity structure. The paper provides comprehensive error analysis, showing a priori estimates and weak-* convergence of the control, with state-convergence rates of and cost-convergence , and demonstrates that the discrete problem has a unique solution in the discretized control space. These results establish rigorous convergence and regularity for numerical approximations of measure-valued, sparsity-promoting parabolic controls, enabling accurate impulse-like actuation in evolving systems.

Abstract

In this paper, we investigate an optimal control problem governed by parabolic equations with measure-valued controls over time. We establish the well-posedness of the optimal control problem and derive the first-order optimality condition using Clarke's subgradients, revealing a sparsity structure in time for the optimal control. Consequently, these optimal control problems represent a generalization of impulse control for evolution equations. To discretize the optimal control problem, we employ the space-time finite element method. Here, the state equation is approximated using piecewise linear and continuous finite elements in space, alongside a Petrov-Galerkin method utilizing piecewise constant trial functions and piecewise linear and continuous test functions in time. The control variable is discretized using the variational discretization concept. For error estimation, we initially derive a priori error estimates and stabilities for the finite element discretizations of the state and adjoint equations. Subsequently, we establish weak-* convergence for the control under the norm , with a convergence order of for the state.
Paper Structure (16 sections, 21 theorems, 170 equations)

This paper contains 16 sections, 21 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations for function spaces
  4. Well-posedness of the state equation
  5. Optimal control problems
  6. Well-posedness of the optimal control problem
  7. First order optimality system
  8. The regularity of solutions
  9. Finite element approximations
  10. Notations for finite element methods
  11. Interpolation and projection operators
  12. Discretization of the optimal control problem
  13. Error estimates for the state and adjoint equations
  14. Stability estimates for the discrete state and adjoint state
  15. A priori error estimates for the state and adjoint equations
  16. ...and 1 more sections

Key Result

lemma 1

For given $g\in L^2(I;H^{-1}(\Omega))$ and $z_T\in L^2(\Omega)$, there exists a unique solution $z\in L^2(I;H^1_0(\Omega))\cap H^1(I;H^{-1}(\Omega))$ to the following problem Moreover, the following estimate holds If, in addition, $g\in L^2(I;L^2(\Omega))\ \hbox{\rm and}\ z_T\in H^1_0(\Omega)$, then $z\in L^2(I;H^2(\Omega)\cap H^1_0(\Omega))\cap\, H^1(I;L^2(\Omega))\cap\,C(\bar{I};H^1_0(\Omega))

Theorems & Definitions (45)

  • Remark 2.1
  • lemma 1
  • proof
  • definition 1
  • theorem 1
  • proof
  • proposition 1
  • proof
  • theorem 2
  • proof
  • ...and 35 more