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Theoretical analysis of a multi-objective controllability problem for the linear wave equation on a non-cylindrical domain

Pedro Paulo A. Oliveira, Isaías P. de Jesus, Gilcenio R. de Sousa-Neto

TL;DR

This work studies a multi-objective Stackelberg boundary control problem for the linear wave equation on a moving domain $Q_{T,\alpha}$, with leader and follower controls acting on two boundary portions. A cylindrical transformation via $y=\frac{x}{\alpha(t)}$ yields a forward–adjoint system with a variable-coefficient operator $L$, enabling an optimality framework that first determines follower controls in feedback form and then solves the leader problem through Fenchel–Rockafellar duality, under time thresholds $T>T_1$ (and $T>T_2$ for the second problem). The main results (Theorems $\['M1'\}$ and $\['M2'\}$) establish the existence of leader controls $f$ (and $g$) that achieve approximate controllability to target states while minimizing respective cost functionals, with explicit expressions $f=-\frac{1}{\alpha^2(t)}\varphi_y$ (and similarly for $g$) where $\varphi$ solves an adjoint equation. The analysis relies on a follower–leader decomposition, an optimality system for the followers, a density argument ensuring range-density, and duality to convert the leader problem into a tractable variational problem. The results generalize prior work from linear boundaries to more general moving-boundary functions and point to extensions to other PDEs and multi-objective equilibria such as Stackelberg–Nash and Pareto approaches.

Abstract

In this article, we investigate certain theoretical aspects of the hierarchical controllability problem in one-dimensional wave equations within a moving domain using Stackelberg strategy. The controls are applied along a portion of the boundary and establish an equilibrium strategy among them, considering a leader control and a follower. We consider a linear wave equation.

Theoretical analysis of a multi-objective controllability problem for the linear wave equation on a non-cylindrical domain

TL;DR

This work studies a multi-objective Stackelberg boundary control problem for the linear wave equation on a moving domain , with leader and follower controls acting on two boundary portions. A cylindrical transformation via yields a forward–adjoint system with a variable-coefficient operator , enabling an optimality framework that first determines follower controls in feedback form and then solves the leader problem through Fenchel–Rockafellar duality, under time thresholds (and for the second problem). The main results (Theorems and ) establish the existence of leader controls (and ) that achieve approximate controllability to target states while minimizing respective cost functionals, with explicit expressions (and similarly for ) where solves an adjoint equation. The analysis relies on a follower–leader decomposition, an optimality system for the followers, a density argument ensuring range-density, and duality to convert the leader problem into a tractable variational problem. The results generalize prior work from linear boundaries to more general moving-boundary functions and point to extensions to other PDEs and multi-objective equilibria such as Stackelberg–Nash and Pareto approaches.

Abstract

In this article, we investigate certain theoretical aspects of the hierarchical controllability problem in one-dimensional wave equations within a moving domain using Stackelberg strategy. The controls are applied along a portion of the boundary and establish an equilibrium strategy among them, considering a leader control and a follower. We consider a linear wave equation.

Paper Structure

This paper contains 10 sections, 3 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Well-posedness of the problem
  3. Caracterization of the problems
  4. Proof of the Theorem \ref{['M1']}
  5. Proof of the Theorem \ref{['M2']}
  6. Additional comments and conclusions
  7. Acknowledgement
  8. Declarations
  9. Conflicts of interest
  10. References

Key Result

Theorem 1.1

Let $u_0\in L^2(0,1)$, $u_1\in H^{-1}(0,1)$, $u^T\in L^2(0,\alpha(T))$, and $u_2\in L^2(Q_{T,\alpha})$. Let us assume that conditions $(H1)-(H3)$ hold true and Then, if $\tilde{\sigma}$ is large enough, for each $\varepsilon>0$ there exist controls $f_\varepsilon\in L^2(\Sigma_T^0)$ and $v_\varepsilon\in L^2(\Sigma_T^0)$ such that

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Theorem 3.1: Optimality system for the follower control