New Stability and Exact Observability Conditions for Semilinear Wave Equations
Provides theoretical guarantees for state estimation in higher-dimensional semilinear wave equations, which is important for control and inverse problems.
The paper extends LMI-based stability and observability conditions from 1-D to n-D semilinear wave equations on hypercubes, providing an upper bound on exact observability time. For 1-D locally Lipschitz nonlinearities, it estimates the region of uniquely recoverable initial conditions.
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.