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Solution stability of parabolic optimal control problems with fixed state-distribution of the controls

Alberto Domínguez Corella, Nicolai Jork, Vladimir M. Veliov

TL;DR

This work develops a rigorous SMsR framework for parabolic optimal control problems with controls that have fixed spatial distributions, enabling stability analysis of the optimality system under data perturbations. By formulating the optimality conditions as a set-valued inclusion 0 in an optimality mapping and proving strong metric subregularity at a reference solution, the authors obtain explicit error bounds linking perturbations in the data to deviations in the state, adjoint state, and control. The approach introduces unified second-order sufficient conditions that blend $L^1$- and $L^2$-type coercivity, yielding growth guarantees and local optimality results that persist under parabolic dynamics and fixed spatial distributions. The SMsR results provide a foundation for stable numerical methods and connect to cone-based reformulations and to a companion paper addressing space-time dependent controls, broadening the toolkit for stability analysis in semilinear parabolic OCPs.

Abstract

The paper presents results about strong metric subregularity of the optimality mapping associated with the system of first-order necessary optimality conditions for a problem of optimal control of a semilinear parabolic equation. The control has a predefined spatial distribution and only the magnitude at any time is a subject of choice. The obtained conditions for subregularity imply, in particular, sufficient optimality conditions that extend the known ones. The paper is complementary to a companion one by the same authors, in which a distributed control is considered.

Solution stability of parabolic optimal control problems with fixed state-distribution of the controls

TL;DR

This work develops a rigorous SMsR framework for parabolic optimal control problems with controls that have fixed spatial distributions, enabling stability analysis of the optimality system under data perturbations. By formulating the optimality conditions as a set-valued inclusion 0 in an optimality mapping and proving strong metric subregularity at a reference solution, the authors obtain explicit error bounds linking perturbations in the data to deviations in the state, adjoint state, and control. The approach introduces unified second-order sufficient conditions that blend - and -type coercivity, yielding growth guarantees and local optimality results that persist under parabolic dynamics and fixed spatial distributions. The SMsR results provide a foundation for stable numerical methods and connect to cone-based reformulations and to a companion paper addressing space-time dependent controls, broadening the toolkit for stability analysis in semilinear parabolic OCPs.

Abstract

The paper presents results about strong metric subregularity of the optimality mapping associated with the system of first-order necessary optimality conditions for a problem of optimal control of a semilinear parabolic equation. The control has a predefined spatial distribution and only the magnitude at any time is a subject of choice. The obtained conditions for subregularity imply, in particular, sufficient optimality conditions that extend the known ones. The paper is complementary to a companion one by the same authors, in which a distributed control is considered.
Paper Structure (7 sections, 20 theorems, 99 equations)

This paper contains 7 sections, 20 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Notations, assumptions, and known facts
  3. The optimal control problem
  4. Sufficient optimality conditions
  5. Strong metric subregularity and auxiliary results
  6. The optimality mapping
  7. Strong metric subregularity: main result

Key Result

Theorem 1

For any $u\in L^2(0,T)^m$ the initial-boundary value problem see1 has a unique weak solution $y_u\in W(0,T)$. If $u\in L^r(0,T)^m$ (see Ern) then $y_u\in W(0,T)\cap L^\infty(Q)$. Moreover, there exists a positive constant $D_{r}$, independent of $u$, $g$, $f$ and $y_0$, such that Finally, if $u_k\rightharpoonup u$ weakly in $L^{r}(Q)$, then

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Lemma 4
  • Remark 1
  • Lemma 5
  • Definition 1
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 16 more