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Hierarchic control for the coupled fourth order parabolic equations

Yating Li, Muming Zhang

Abstract

In this paper, we obtain a null controllability result for a coupled fourth order parabolic system based on the Stackelberg-Nash strategies. For this purpose, we first prove the existence and uniqueness of Nash equilibrium pair of the original system and its explicit expression is provided. Next, we investigate the null controllability of Nash equilibrium to the corresponding optimal system. By duality theory, we establish an observability estimate for the coupled fourth order parabolic system. Such an estimate is obtained by a new global Carleman estimate we derived.

Hierarchic control for the coupled fourth order parabolic equations

Abstract

In this paper, we obtain a null controllability result for a coupled fourth order parabolic system based on the Stackelberg-Nash strategies. For this purpose, we first prove the existence and uniqueness of Nash equilibrium pair of the original system and its explicit expression is provided. Next, we investigate the null controllability of Nash equilibrium to the corresponding optimal system. By duality theory, we establish an observability estimate for the coupled fourth order parabolic system. Such an estimate is obtained by a new global Carleman estimate we derived.
Paper Structure (7 sections, 6 theorems, 117 equations)

This paper contains 7 sections, 6 theorems, 117 equations.

Table of Contents

  1. Introduction and main result
  2. Nash equilibrium
  3. Existence and uniqueness of a Nash equilibrium
  4. Explicit expression of the Nash equilibrium pair
  5. Reduction of the null controllability problem
  6. Carleman estimate of the coupled fourth order parabolic system
  7. Observability of the coupled fourth order parabolic system

Key Result

Theorem 1.1

Assume that $O_{1,d}=O_{2,d}$, denoted as $O_{d}$, $O_{d}\cap\omega\neq\emptyset$ and $\mu_{i}$ (i=1,2) are sufficiently large. If and then for any $y^{0}=(y_{1}^{0},y_{2}^{0})\in L^{2}(\Omega)^{2}$, there exists a control $\bar{g}\in L^{2}(\omega\times(0,T))$ and the corresponding Nash equilibrium $(\bar{h}_{1}(\bar{g}),\bar{h}_{2}(\bar{g}))$ such that the solution of (1.1) satisfies (1.6) and

Theorems & Definitions (7)

  • Definition 1.1
  • Theorem 1.1
  • Proposition 2.1
  • Proposition 3.1
  • Lemma 4.1
  • Proposition 4.1
  • Proposition 5.1