New Stability and Exact Observability Conditions for Semilinear Wave Equations

TL;DR

This paper extends stability and observability conditions for wave equations from 1-D to n-D, providing new LMI-based criteria and bounds, with numerical validation for locally Lipschitz nonlinearities.

Contribution

It generalizes existing 1-D results to n-D wave equations, introducing new LMI-based stability conditions and observability bounds, and estimates initial condition recovery regions.

Findings

New LMI-based exponential stability conditions for n-D wave equations

Upper bounds on exact observability time in terms of LMIs

Numerical examples demonstrating effectiveness of the results

Abstract

The problem of estimating the initial state of 1-D wave equations with globally Lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers, and deriving the upper bound for exact observability time in terms of Linear Matrix Inequalities (LMIs) [5]. In the present paper, we generalize this result to n-D wave equations on a hypercube. This extension includes new LMI-based exponential stability conditions for n-D wave equations, as well as an upper bound on the minimum exact observability time in terms of LMIs. For 1-D wave equations with locally Lipschitz nonlinearities, we find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements. The efficiency of the results is illustrated by numerical examples.