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Controllability for a One-Dimensional Wave Equation in a Non-cylindrical Domain

Isaías Pereira de Jesus

Abstract

This paper deals with the controllability for a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. This equation models small vibrations of a string where an endpoint is fixed and the other is moving. As usual, we consider one main control (the leader) and an additional secondary control (the follower). We use Stackelberg-Nash strategies.

Controllability for a One-Dimensional Wave Equation in a Non-cylindrical Domain

Abstract

This paper deals with the controllability for a one-dimensional wave equation with mixed boundary conditions in a non-cylindrical domain. This equation models small vibrations of a string where an endpoint is fixed and the other is moving. As usual, we consider one main control (the leader) and an additional secondary control (the follower). We use Stackelberg-Nash strategies.

Paper Structure

This paper contains 9 sections, 3 theorems, 68 equations.

Table of Contents

  1. Introduction and main result
  2. Statement of the problem
  3. Reduction to controllability problem in a cylindrical domain
  4. Nash equilibrium
  5. Approximate controllability
  6. The optimality system
  7. Some additional comments and questions
  8. Acknowledgements
  9. References

Key Result

Theorem 1.1

Assume that hT and hT10 hold. Let us consider $w_1 \in L^2(\Sigma_1)$ and $w_2$ a Nash equilibrium in the sense soncil. Then $\left(v(T), v'(T)\right)=\left(v(., T, {w_1}, w_2), v'(., T, {w_1}, w_2)\right)$, where $v$ solves the system eq1.14, generates a dense subset of $L^2(\Omega)\times H^{-1}(\O

Theorems & Definitions (6)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.1
  • Theorem 2.1
  • Theorem 4.1