Boundary control of multi-dimensional discrete-velocity kinetic models

TL;DR

The work addresses boundary stabilization of multi-dimensional discrete-velocity kinetic models by exploiting a physically meaningful dissipation structure encoded in the structural stability condition. It constructs a modified Lyapunov functional $L(t)$ and derives boundary control laws that guarantee exponential decay of the deviation from a uniform steady state. The main contributions include a general design rule that yields infinitely many admissible local or nonlocal boundary conditions, and a concrete application to the 2-D coplanar model with demonstrable exponential decay via numerical simulations. The results provide a principled framework for boundary control in semi-linear hyperbolic relaxation systems and point to future extensions to nonuniform steady states and nonlinear regimes.

Abstract

This technical note is concerned with boundary stabilization of multi-dimensional discrete-velocity kinetic models. By exploiting a certain stability structure of the models and adapting an appropriate Lyapunov functional, we derive feasible control laws so that the corresponding solutions decay exponentially in time. The result is illustrated with an application to the two-dimensional coplanar model in a square container. The effectiveness of the derived control laws is confirmed by numerical simulations.

Figures (1)

• Figure 1: Time evolution of logarithm of $L^2$-norm of the solution when applying the derived control laws.

Theorems & Definitions (16)

• Lemma 1
• proof
• Remark 1
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 2
• proof : Proof of Theorem \ref{['T3.1']}