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Approximate controllability for a one-dimensional wave equation with the fixed endpoint control

Isaías Pereira de Jesus

TL;DR

The paper addresses approximate controllability of a one-dimensional wave equation on a moving-boundary non-cylindrical domain. It adopts a Stackelberg leader–follower framework, yielding a Nash equilibrium for the follower and an adjoint-based optimality system; the follower control admits the explicit representation $\widetilde{w}_2=\frac{1}{\widetilde{\sigma}}p_x$ on the follower boundary. Via a Fenchel–Rockafellar duality approach, it derives a leader optimality condition $\widetilde{w}_1=-\varphi_x$ on the leader boundary and couples forward, adjoint, and follower states in the optimality system. Under $0<k<1$ and a sufficiently large final time $T$, the final-state map with the Nash follower has dense range in $L^2(\\Omega_t)\times H^{-1}(\\Omega_t)$, establishing approximate controllability. These results extend controllability analysis to non-cylindrical domains and integrate hierarchical Stackelberg strategies with PDE control.

Abstract

This paper is devoted to the study of the approximate controllability for a one-dimensional wave equation in domains with moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.

Approximate controllability for a one-dimensional wave equation with the fixed endpoint control

TL;DR

The paper addresses approximate controllability of a one-dimensional wave equation on a moving-boundary non-cylindrical domain. It adopts a Stackelberg leader–follower framework, yielding a Nash equilibrium for the follower and an adjoint-based optimality system; the follower control admits the explicit representation on the follower boundary. Via a Fenchel–Rockafellar duality approach, it derives a leader optimality condition on the leader boundary and couples forward, adjoint, and follower states in the optimality system. Under and a sufficiently large final time , the final-state map with the Nash follower has dense range in , establishing approximate controllability. These results extend controllability analysis to non-cylindrical domains and integrate hierarchical Stackelberg strategies with PDE control.

Abstract

This paper is devoted to the study of the approximate controllability for a one-dimensional wave equation in domains with moving boundary. This equation models the motion of a string where an endpoint is fixed and the other one is moving. When the speed of the moving endpoint is less than the characteristic speed, the controllability of this equation is established. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.
Paper Structure (8 sections, 3 theorems, 61 equations)

This paper contains 8 sections, 3 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Problem formulation
  4. Nash equilibrium
  5. On the approximate controllability
  6. Optimality system for the leader
  7. Acknowledgements
  8. References

Key Result

Theorem 3.1

For each $\widetilde{w}_1 \in L^2(\Sigma_1)$ there exists a unique Nash equilibrium $\widetilde{w}_2$ in the sense of son. Moreover, the follower $\widetilde{w}_2$ is given by where $\{ v,p \}$ is the unique solution of (the optimality system) Of course, $\{ u,p \}$ depends on $\widetilde{w}_1$:

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Theorem 4.1
  • Definition 5.1
  • Theorem 5.1
  • Remark 5.1