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Locally Lipschitz stability of solutions to a parametric parabolic optimal control problem with mixed pointwise constraints

Huynh Khanh

Abstract

A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. By analyzing regularity and establishing stability condition of Lagrange multipliers we prove that, if the strictly second-order sufficient condition for the unperturbed problem is valid, then the solutions of the problems as well as the associated Lagrange multipliers are locally Lipschitz continuous functions of parameters.

Locally Lipschitz stability of solutions to a parametric parabolic optimal control problem with mixed pointwise constraints

Abstract

A class of parametric optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is investigated. The perturbations appear in the objective functional, the state equation and in mixed pointwise constraints. By analyzing regularity and establishing stability condition of Lagrange multipliers we prove that, if the strictly second-order sufficient condition for the unperturbed problem is valid, then the solutions of the problems as well as the associated Lagrange multipliers are locally Lipschitz continuous functions of parameters.
Paper Structure (7 sections, 13 theorems, 187 equations)

This paper contains 7 sections, 13 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Assumptions and statement of the main result
  3. Analysis of the state equation
  4. First-order optimality conditions and regularity of Lagrange multipliers
  5. Stability of Lagrange multipliers
  6. Second-order sufficient conditions for the unperturbed problem
  7. Proof of the main result

Key Result

Theorem 2.1

Let $\overline z = (\overline y, \overline u) \in \Phi(\overline w)$. Suppose that assumptions $(H1)$-$(H4)$ are valid and there exists a couple $(\varphi, e) \in \Lambda_\infty [\overline z, \overline w]$ such that the following strictly second-order condition: is satisfied for all $(y, u)\in\mathcal{C}_2[(\overline y, \overline u), \overline w] \setminus \{(0,0)\}$. Furthermore, there exists a

Theorems & Definitions (18)

  • Example 2.1
  • Example 2.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 4.1
  • Lemma 4.1
  • ...and 8 more