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Regularity of multipliers and second-order optimality conditions for semilinear parabolic optimal control problems with mixed pointwise constraints

Huynh Khanh, Bui Trong Kien

Abstract

A class of optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is considered. We give some criteria under which the first and second-order optimality conditions are of KKT-type. We then prove that the Lagrange multipliers belong to $L^p$-spaces. Moreover, we show that if the initial value is good enough and boundary $\partialΩ$ has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.

Regularity of multipliers and second-order optimality conditions for semilinear parabolic optimal control problems with mixed pointwise constraints

Abstract

A class of optimal control problems governed by semilinear parabolic equations with mixed pointwise constraints is considered. We give some criteria under which the first and second-order optimality conditions are of KKT-type. We then prove that the Lagrange multipliers belong to -spaces. Moreover, we show that if the initial value is good enough and boundary has a property of positive geometric density, then multipliers and optimal solutions are Hölder continuous.
Paper Structure (6 sections, 6 theorems, 159 equations)

This paper contains 6 sections, 6 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Some auxiliary facts
  3. State equation and linearized equation
  4. A specific mathematical programming problem
  5. Existence of regular multipliers in second-order optimality conditions
  6. Hölder continuity of multipliers and optimal solutions

Key Result

Lemma 2.1

Suppose that $(H1)$ and $(H2)$ are satisfied and $y_0\in L^\infty(\Omega)\cap H_0^1(\Omega)$. Then for each $u\in L^p(0, T; H)$ with $p>\frac{4}{4-N}$, the state equation P2-P3 has a unique solution $y\in Y$ and there exist positive constants $C_1>0$ and $C_2>0$ such that and

Theorems & Definitions (8)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Example 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.1