# Event-triggered boundary damping of a linear wave equation

## TL;DR

This paper analyzes the stabilization of a multidimensional wave equation using an event-triggered boundary control mechanism, ensuring convergence and avoiding Zeno behavior through Lyapunov-based conditions.

## Contribution

It introduces a novel event-triggered boundary damping strategy for wave equations, with proven existence, regularity, and convergence of solutions.

## Key findings

- Solutions converge into a tunable compact set containing the origin.
- The event-triggering mechanism prevents Zeno behavior.
- Sufficient Lyapunov-based conditions ensure stability.

## Abstract

This article presents an analysis of the stabilization of a multidimensional partial differential wave equation under a well designed event-triggering mechanism that samples the boundary control input. The wave equation is set in a bounded domain and the control is performed through a boundary classical damping term, where the Neumann boundary condition is made proportional to the velocity. First of all, existence and regularity of the solution to the closed-loop system under the event-triggering mechanism of the control are proven. Then, sufficient conditions based on the use of a specific Lyapunov functional are proposed in order to ensure that the solutions converge into a compact set containing the origin, that can be tuned by the designer. Furthermore, as expected, any Zeno behavior of the closed-loop system is avoided.