Stabilization of the wave equation with boundary or internal distributed delay

By Serge Nicaise, Cristina Pignotti. Differential and Integral Equations, 2008

366 citations recorded by the source corpus.

We consider the wave equation in a bounded region with a smooth boundary with distributed delay on the boundary or into the domain. In both cases, under suitable assumptions, we prove the expo- nential stability of the solution. These results are obtained by introduc- ing suitable energies and by proving some observability inequalities. For an internal distributed delay, we further show some instability results.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b0c77e1a54fadc868100e7339cbad409e49eb606

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In Computational Theory and Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback (2012, background): “For example, it was proved in [9, 10, 21, 22, 30] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay unless additional conditions or control terms have been used.”
Existence and asymptotic stability of a viscoelastic wave equation with a delay (2011, background): “In order to prove the existence of a unique solution of problem (1), we introduce as in [ 18 ], the new variable”
A Blow-Up Result in a Nonlinear Wave Equation with Delay (2016, result): “exponential stability results were obtained in [21] under the condition ∫ τ2”