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Remarks on hierarchic control for the wave equation in a non cylindrical domain

Isaias Pereira de Jesus

TL;DR

This work develops a hierarchic Stackelberg framework for the wave equation in a time-dependent, non-cylindrical domain. By decomposing boundary controls into a leader and a follower, it proves the existence and uniqueness of a Nash equilibrium for the follower and establishes approximate controllability of the state with respect to the leader, using a cylindrical-domain reformulation and adjoint-based analysis. The leader's optimality system is derived via Fenchel-Rockafellar duality, yielding an explicit gradient condition $w_1= -\\frac{1}{k^2(t)}\\varphi_\\nu$ on the leader boundary, where $\\varphi$ solves a coupled adjoint problem. Together, these results extend Stackelberg/Nash-type controllability to moving-domain wave equations and provide a rigorous variational framework for the optimal leader control.

Abstract

In this paper we establish hierarchic control for the wave equation in a non cylindrical domain $\widehat{Q}$ of $\mathbb{R}^{n + 1}$. We assume that we can act in the dynamic of the system by a hierarchy of controls. According to the formulation given by H. Von Stackelberg \cite{S}, there are local controls, called followers and global controls, called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as "the leader" and the other one as "the follower". This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.

Remarks on hierarchic control for the wave equation in a non cylindrical domain

TL;DR

This work develops a hierarchic Stackelberg framework for the wave equation in a time-dependent, non-cylindrical domain. By decomposing boundary controls into a leader and a follower, it proves the existence and uniqueness of a Nash equilibrium for the follower and establishes approximate controllability of the state with respect to the leader, using a cylindrical-domain reformulation and adjoint-based analysis. The leader's optimality system is derived via Fenchel-Rockafellar duality, yielding an explicit gradient condition on the leader boundary, where solves a coupled adjoint problem. Together, these results extend Stackelberg/Nash-type controllability to moving-domain wave equations and provide a rigorous variational framework for the optimal leader control.

Abstract

In this paper we establish hierarchic control for the wave equation in a non cylindrical domain of . We assume that we can act in the dynamic of the system by a hierarchy of controls. According to the formulation given by H. Von Stackelberg \cite{S}, there are local controls, called followers and global controls, called leaders. In fact, one considers situations where there are two cost (objective) functions. One possible way is to cut the control into two parts, one being thought of as "the leader" and the other one as "the follower". This situation is studied in the paper, with one of the cost functions being of the controllability type. Existence and uniqueness is proven. The optimality system is given in the paper.
Paper Structure (5 sections, 3 theorems, 68 equations)

This paper contains 5 sections, 3 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Nash equilibrium
  4. On the approximate controllability
  5. Optimality system for the leader

Key Result

Theorem 1

For each $w_1 \in L^2(\Sigma_1)$ there exists an unique Nash equilibrium $w_2$ in the sense of (soncil). Moreover, the follower $w_2$ is given by where $\{ v,p \}$ is the unique solution of (the optimality system) Of course $\{ v,p \}$ depends on $w_1$:

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Theorem 2
  • Theorem 3