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Control and stabilization of cascade coupled systems: application to a 1-d heat and wave coupled system

Lucas Davron, Pierre Lissy, Swann Marx

Abstract

We study cascade coupled systems, for which our prototypical example is a 1-d heat equation coupled with a 1-d wave equation. The heat component is controlled through one boundary and the information is transmitted through another one to the wave component, while the wave component does not influence the heat component. Our aim is to understand the well-posedness, controllability and stabilizability properties for such a system. Establishing well-posedness is tedious using the classical energy method, which motivates us to take advantage of the cascade structure. Taking again advantage of this structure, we prove a simultaneous exact and approximate controllability result. Finally, we obtain polynomial stabilization by means of a closed-loop control defined through the solution to a Sylvester equation. These results are all discussed in an abstract LTI framework and most of our findings apply to more general situations.

Control and stabilization of cascade coupled systems: application to a 1-d heat and wave coupled system

Abstract

We study cascade coupled systems, for which our prototypical example is a 1-d heat equation coupled with a 1-d wave equation. The heat component is controlled through one boundary and the information is transmitted through another one to the wave component, while the wave component does not influence the heat component. Our aim is to understand the well-posedness, controllability and stabilizability properties for such a system. Establishing well-posedness is tedious using the classical energy method, which motivates us to take advantage of the cascade structure. Taking again advantage of this structure, we prove a simultaneous exact and approximate controllability result. Finally, we obtain polynomial stabilization by means of a closed-loop control defined through the solution to a Sylvester equation. These results are all discussed in an abstract LTI framework and most of our findings apply to more general situations.
Paper Structure (15 sections, 15 theorems, 223 equations)

This paper contains 15 sections, 15 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Controllability
  4. Stabilization
  5. Considerations on abstract cascade coupled systems
  6. Abstract linear systems
  7. Cascade coupled systems
  8. Riesz basis property
  9. Applications to the heat-wave system
  10. Polynomial stabilization
  11. The Sylvester equation
  12. Collocated feedback
  13. Asymptotic stability
  14. A polynomial rate
  15. Proof of Proposition \ref{['prop:zz_pour_nous']}

Key Result

Proposition 1.1

The operator $\mathcal{A}$ generates a $C_0$-semigroup on $\mathcal{X}$, and $\mathcal{B}$ is an admissible control operator for the semigroup generated by $\mathcal{A}$.

Theorems & Definitions (38)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 28 more