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Augmented Lagrange method for optimal control problems of parabolic equation with state constraints

Weilong You, Fu Zhang

TL;DR

The paper tackles optimal control of parabolic PDEs with pointwise state constraints by developing an augmented Lagrange framework. It formulates an augmented Lagrangian that enforces $y\le\psi$ and solves ensuing subproblems with the Method of Successive Approximations guided by Pontryagin's Maximum Principle, establishing strong convergence of the primal variables and weak convergence of duals. Key contributions include a rigorous convergence analysis for the augmented-Lagrangian algorithm, a detailed multiplier update strategy, and demonstration that the method converges to the original constrained problem without requiring Slater-like regularization. The numerical experiments corroborate theoretical results, showing $R^+_n\to 0$ and near-analytic-state accuracy, highlighting the method's practicality for state-constrained parabolic control problems.

Abstract

The augmented Lagrange method is employed to address the optimal control problem involving pointwise state constraints in parabolic equations. The strong convergence of the primal variables and the weak convergence of the dual variables are rigorously established. The sub-problems arising in the algorithm are solved using the Method of Successive Approximations (MSA), derived from Pontryagin's principle. Numerical experiments are provided to validate the convergence of the proposed algorithm.

Augmented Lagrange method for optimal control problems of parabolic equation with state constraints

TL;DR

The paper tackles optimal control of parabolic PDEs with pointwise state constraints by developing an augmented Lagrange framework. It formulates an augmented Lagrangian that enforces and solves ensuing subproblems with the Method of Successive Approximations guided by Pontryagin's Maximum Principle, establishing strong convergence of the primal variables and weak convergence of duals. Key contributions include a rigorous convergence analysis for the augmented-Lagrangian algorithm, a detailed multiplier update strategy, and demonstration that the method converges to the original constrained problem without requiring Slater-like regularization. The numerical experiments corroborate theoretical results, showing and near-analytic-state accuracy, highlighting the method's practicality for state-constrained parabolic control problems.

Abstract

The augmented Lagrange method is employed to address the optimal control problem involving pointwise state constraints in parabolic equations. The strong convergence of the primal variables and the weak convergence of the dual variables are rigorously established. The sub-problems arising in the algorithm are solved using the Method of Successive Approximations (MSA), derived from Pontryagin's principle. Numerical experiments are provided to validate the convergence of the proposed algorithm.

Paper Structure

This paper contains 16 sections, 15 theorems, 83 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Setting of the control problem
  4. Existence of solutions
  5. Optimality conditions
  6. The augmented Lagrange method
  7. The augmented Lagrange optimal control problem
  8. The augmented Lagrange algorithm in brief
  9. The multiplier update rule
  10. The augmented Lagrange algorithm in detail
  11. Convergence of algorithm
  12. Optimization of sub-problems
  13. Pontryagin's Maximum Principle
  14. Method of Successive Approximations
  15. Numerical tests
  16. ...and 1 more sections

Key Result

Lemma 2.1

(Existence of weak solution)When the above assumptions fulfilled, for every $u \in L^r(\Omega_T),\; v \in L^s(\Sigma_T)$, there exists a weak solution $y \in W(0,T;L^2(\Omega),H^1(\Omega))$ of the state equation.Evans2010P

Figures (3)

  • Figure 1: Computed discrete optimal state y (right) and optimal control u (left)
  • Figure 2: The number of successes n, the corresponding index $R^+_n$ (left), and the growth trend of penalty factor $\rho_k$ (right)
  • Figure 3: Computed discrete optimal state at time $T$ (left), Theoretical optimal state at time $T$ (center), Difference between computed and theoretical optimal state at time $T$ (right)

Theorems & Definitions (31)

  • Definition 1
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 2
  • Theorem 2.5
  • ...and 21 more