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New second order sufficient optimality conditions for state constrained parabolic control problems

Eduardo Casas, Mariano Mateos, Arnd Rösch

Abstract

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of $L^2(Q)$. Our results are valid for spatial domains of dimension less than or equal to three.

New second order sufficient optimality conditions for state constrained parabolic control problems

Abstract

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of . Our results are valid for spatial domains of dimension less than or equal to three.
Paper Structure (5 sections, 12 theorems, 95 equations)

This paper contains 5 sections, 12 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Assumptions and study of the equations and the functional
  3. First order necessary optimality conditions
  4. Sufficient optimality conditions
  5. Bilateral constraints

Key Result

Theorem 2.1

For every $u\in L^{p}(Q)$ there exists a unique solution $y_u$ of E01, belonging to the space $L^2(0,T;H^1_0(\Omega))\cap C(\bar{Q})$. Moreover, there exist a monotone non-decreasing function $\eta:[0,\infty)\to [0,+\infty)$ and a positive constant $K$ such that Moreover, for all $R>0$ there exists $C_{p,R}>0$ such that $\|y_u-y_{\bar{u}}\|_{C(\bar{Q})}\leq C_{p,R}\|u-\bar{u}\|_{L^{p}(Q)}$ for a

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 1
  • Corollary 2.6
  • Corollary 2.7
  • ...and 10 more