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Controllability and Inverse Problems for Hyperbolic and Dispersive Equations with Dynamic Boundary Conditions

S. E. Chorfi, L. Maniar, R. Morales

TL;DR

The paper addresses controllability and inverse problems for hyperbolic and dispersive PDEs with dynamic boundary conditions, leveraging Carleman estimates tailored to boundary dynamics. For waves, exact boundary controllability is established under δ > d and a minimal time threshold T_*, with an adjoint observability inequality underpinning the result and explicit formulae for T_*. For Schrödinger equations with dynamic boundaries, exact controllability is obtained under δ > d, supported by Carleman-based observability, and Lipschitz stability results are proved for inverse source and coefficient problems, together with a Carleman-based reconstruction approach. The work also discusses limitations in the δ ≤ d regime and emphasizes geometric and nonlinear extensions as important future directions. Overall, the results advance controllability and stable inversion in systems with time-varying boundary interactions, with potential applications to wave/quantum devices where boundary dynamics are critical.

Abstract

This review examines classical and recent results on controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions. We aim to illustrate the applicability of Carleman estimates to establish exact controllability of such equations and derive Lipschitz stability estimates for inverse problems of source terms and coefficients with general dynamic boundary conditions. We highlight the challenges associated with dynamic boundary conditions compared to classical static ones. Finally, we conclude with a discussion of open problems and future research directions.

Controllability and Inverse Problems for Hyperbolic and Dispersive Equations with Dynamic Boundary Conditions

TL;DR

The paper addresses controllability and inverse problems for hyperbolic and dispersive PDEs with dynamic boundary conditions, leveraging Carleman estimates tailored to boundary dynamics. For waves, exact boundary controllability is established under δ > d and a minimal time threshold T_*, with an adjoint observability inequality underpinning the result and explicit formulae for T_*. For Schrödinger equations with dynamic boundaries, exact controllability is obtained under δ > d, supported by Carleman-based observability, and Lipschitz stability results are proved for inverse source and coefficient problems, together with a Carleman-based reconstruction approach. The work also discusses limitations in the δ ≤ d regime and emphasizes geometric and nonlinear extensions as important future directions. Overall, the results advance controllability and stable inversion in systems with time-varying boundary interactions, with potential applications to wave/quantum devices where boundary dynamics are critical.

Abstract

This review examines classical and recent results on controllability and inverse problems for hyperbolic and dispersive equations with dynamic boundary conditions. We aim to illustrate the applicability of Carleman estimates to establish exact controllability of such equations and derive Lipschitz stability estimates for inverse problems of source terms and coefficients with general dynamic boundary conditions. We highlight the challenges associated with dynamic boundary conditions compared to classical static ones. Finally, we conclude with a discussion of open problems and future research directions.

Paper Structure

This paper contains 17 sections, 9 theorems, 63 equations, 4 figures.

Table of Contents

  1. Introduction and motivation
  2. Literature on hyperbolic equations
  3. Literature on dispersive equations
  4. Outline of the paper
  5. Preliminaries
  6. Geometric setting
  7. Functional setting
  8. Hyperbolic equations with dynamic boundary conditions
  9. Boundary controllability
  10. Inverse problems
  11. Dispersive equations with dynamic boundary conditions
  12. Boundary controllability
  13. Inverse problems
  14. Conclusion and open problems
  15. The case $\delta \leq d$
  16. ...and 2 more sections

Key Result

Lemma 1

The weight function $\psi_{0}$ satisfies the following properties:

Figures (4)

  • Figure 1: Domain satisfying GCC.
  • Figure 2: Domain with an interior observation region which does not satisfy GCC.
  • Figure 3: Domain with a boundary observation region which does not satisfy GCC.
  • Figure 4: Domain satisfying the geometric setting.

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • ...and 2 more