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Pontryagin's Principle Based Algorithms for Optimal Control Problems of Parabolic Equation

Weilong You, Fu Zhang

TL;DR

The paper tackles optimal control of semilinear parabolic equations with state constraints by leveraging Pontryagin's maximum principle within a Method of Successive Approximations (MSA) framework. It develops an augmented MSA (AMSA) using an augmented Lagrangian penalty to enforce stable, guaranteed descent, and provides error estimates for first/second derivatives of the reduced objective under regularity assumptions. Theoretical results establish convergence of AMSA in $L^2$ for the controls when the penalty parameter satisfies $\rho > \tilde{C}$, supported by a numerical demonstration on a Neumann-boundary parabolic problem. The work delivers a rigorous, implementable approach for state-constrained parabolic optimal control with provable descent and practical discretization, highlighting both its robustness and potential to converge to local optima due to gradient-based subproblem solves.

Abstract

This paper applies the Method of Successive Approximations (MSA) based on Pontryagin's principle to solve optimal control problems with state constraints for semilinear parabolic equations. Error estimates for the first and second derivatives of the function are derived under \( L^{\infty} \)-bounded conditions. An augmented MSA is developed using the augmented Lagrangian method, and its convergence is proven. The effectiveness of the proposed method is demonstrated through numerical experiments.

Pontryagin's Principle Based Algorithms for Optimal Control Problems of Parabolic Equation

TL;DR

The paper tackles optimal control of semilinear parabolic equations with state constraints by leveraging Pontryagin's maximum principle within a Method of Successive Approximations (MSA) framework. It develops an augmented MSA (AMSA) using an augmented Lagrangian penalty to enforce stable, guaranteed descent, and provides error estimates for first/second derivatives of the reduced objective under regularity assumptions. Theoretical results establish convergence of AMSA in for the controls when the penalty parameter satisfies , supported by a numerical demonstration on a Neumann-boundary parabolic problem. The work delivers a rigorous, implementable approach for state-constrained parabolic optimal control with provable descent and practical discretization, highlighting both its robustness and potential to converge to local optima due to gradient-based subproblem solves.

Abstract

This paper applies the Method of Successive Approximations (MSA) based on Pontryagin's principle to solve optimal control problems with state constraints for semilinear parabolic equations. Error estimates for the first and second derivatives of the function are derived under -bounded conditions. An augmented MSA is developed using the augmented Lagrangian method, and its convergence is proven. The effectiveness of the proposed method is demonstrated through numerical experiments.
Paper Structure (10 sections, 6 theorems, 56 equations, 2 figures, 2 algorithms)

This paper contains 10 sections, 6 theorems, 56 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Problem setting
  4. Pontryagin's Principle
  5. Method of Successive Approximations
  6. Basic MSA
  7. Error Estimate
  8. Augmented MSA
  9. Convergence of algorithm
  10. Numerical tests

Key Result

Theorem 2.1

Let $(\bar{y},\bar{u},\bar{v})$ be the solution of $(P)$. Then there exists a unique adjoint state $\bar{p} \in \mathcal{W}(0,T;L^2(\Omega),H^{1}(\Omega))\cap C(\Omega_T)$, satisfying the following equation. and such that

Figures (2)

  • Figure 1: The difference in the objective function $J(u_{i+1})-J(u_i)$
  • Figure 2: Computed discrete optimal state $y$ (right) and optimal control $u$ (left)

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 2 more