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On the relation between approaches for boundary feedback control of hyperbolic systems

Michael Herty, Ferdinand Thein

TL;DR

The paper addresses boundary feedback stabilization of multi-dimensional hyperbolic balance laws by linking two prominent approaches: the SSC-based framework of Yang 2024 and the LMI-based method of Herty 2023a. It proves that any SSC system satisfies a feasible Lyapunov potential $\mu(\mathbf{x})$ making the LMI $\mathbf{A}(\mathbf{m}) \le 0$ hold, yielding exponential decay of the Lyapunov function, and demonstrates this connection through a detailed Saint-Venant example. The work highlights how appropriate weight functions and boundary conditions enable exponential stability and shows that the LMI approach extends to non-SSC cases (via Herty 2023a) as illustrated by a diagonal-system example. Numerical results with MUSCL schemes validate the theory, showing decay rates that align with, and can exceed, theoretical predictions, underscoring the practical relevance for complex multi-D hyperbolic PDEs and networks.

Abstract

Stabilization of partial differential equations is a topic of utmost importance in mathematics as well as in engineering sciences. Concerning one dimensional problems there exists a well developed theory. Due to numerous important applications the interest in boundary feedback control of multi-dimensional hyperbolic systems is increasing. In the present work we want to discuss the relation between some of the most recent results available in the literature. The key result of the present work is to show that the type of system discussed in Yang and Yong (2024) identifies a particular class which falls into the framework presented in Herty and Thein (2024).

On the relation between approaches for boundary feedback control of hyperbolic systems

TL;DR

The paper addresses boundary feedback stabilization of multi-dimensional hyperbolic balance laws by linking two prominent approaches: the SSC-based framework of Yang 2024 and the LMI-based method of Herty 2023a. It proves that any SSC system satisfies a feasible Lyapunov potential making the LMI hold, yielding exponential decay of the Lyapunov function, and demonstrates this connection through a detailed Saint-Venant example. The work highlights how appropriate weight functions and boundary conditions enable exponential stability and shows that the LMI approach extends to non-SSC cases (via Herty 2023a) as illustrated by a diagonal-system example. Numerical results with MUSCL schemes validate the theory, showing decay rates that align with, and can exceed, theoretical predictions, underscoring the practical relevance for complex multi-D hyperbolic PDEs and networks.

Abstract

Stabilization of partial differential equations is a topic of utmost importance in mathematics as well as in engineering sciences. Concerning one dimensional problems there exists a well developed theory. Due to numerous important applications the interest in boundary feedback control of multi-dimensional hyperbolic systems is increasing. In the present work we want to discuss the relation between some of the most recent results available in the literature. The key result of the present work is to show that the type of system discussed in Yang and Yong (2024) identifies a particular class which falls into the framework presented in Herty and Thein (2024).
Paper Structure (13 sections, 1 theorem, 90 equations, 2 figures)

This paper contains 13 sections, 1 theorem, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The generic LMI approach
  3. The approach for SSC systems
  4. Main result
  5. Examples
  6. Application to the Saint-Venant system
  7. The case of a diagonal system
  8. Numerical Results
  9. The Saint-Venant system
  10. The diagonal system
  11. Summary
  12. Eigenstructure of the Saint-Venant Equations
  13. Boundary Control

Key Result

Theorem 4.1

Let system eq:hyp_cons_sys2 be given and assume it satisfies the SSC property, i.e. the properties (i) - (iii) hold. Then there exists a feasible Lyapunov potential $\mu(\mathbf{x})$ such that the LMI ineq:control_LMI2 holds and the Lyapunov function eq:lyapunov_gen decays exponentially.

Figures (2)

  • Figure 1: Numerical results at $t_{end} = 3$: The solutions for the components a) $\tilde{h}$, b) $w$ and c) $v$ are given. The computed decay rates for both Lyapunov functions (blue Yang2024 and red Herty2023a) obtained with the MUSCL-FV scheme are given and compared to the decay rate $C = 1.4$ (purple) which is compliant with \ref{['constr_decay_rate']}, see d).
  • Figure 2: Numerical results at $t_{end} = 3$: The solutions for the components a) $w_1$, b) $w_2$ and c) $w_3$ are given. The computed (blue) decay rate for the Lyapunov function obtained with the MUSCL-FV scheme is given and compared to the theoretical decay rate (red), see d).

Theorems & Definitions (4)

  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4