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On the solution stability of parabolic optimal control problems.

Alberto Domínguez Corella1, Nicolai Jork1, Vladimir M Veliov1

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This study analyzes the stability of optimal control problems governed by semilinear parabolic partial differential equations. It establishes Hölder or Lipschitz stability for optimal solutions under perturbations, enhancing understanding of control system robustness.

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Area of Science:

  • Optimal Control Theory
  • Partial Differential Equations (PDEs)
  • Mathematical Optimization

Background:

  • Investigates stability of solutions for optimal control problems (OCPs).
  • Focuses on OCPs constrained by semilinear parabolic partial differential equations.
  • Examines dependence of optimal solutions on perturbations in equations and objective functionals.

Purpose of the Study:

  • To obtain Hölder or Lipschitz stability results for optimal solutions.
  • To analyze stability under nonlinear state and control variable perturbations.
  • To establish metric subregularity of the mapping associated with optimality conditions.

Main Methods:

  • Extends recent assumptions on joint growth of first and second variations of the objective functional.
  • Utilizes the concept of metric subregularity for the mapping of first-order necessary optimality conditions.
  • Applies these methods to analyze stability of optimal control problems with semilinear parabolic PDEs.

Main Results:

  • Achieves Hölder or Lipschitz dependence of the optimal solution on perturbations.
  • Demonstrates metric subregularity of the mapping associated with optimality conditions.
  • Obtains a Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter.

Conclusions:

  • The study provides theoretical guarantees for the stability of optimal control solutions.
  • Metric subregularity is a key property enabling stability analysis and error estimates.
  • Results have implications for the reliability of approximation methods in optimal control.