Stabilization of se ond order evolution equations
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Stabilization of se ond order evolution equations with unbounded feedba k with time-dependent delay Emilia Fridman ? , Serge Ni aise † , Julie Valein ‡ Mar h 24, 2009 Abstra t We onsider abstra t se ond order evolution equations with unbounded feedba k with time-varying delay. Existen e results are obtained under some realisti assumptions. We prove the exponential de ay under some onditions by introdu ing an abstra t Lyapunov fun tional. Our abstra t framework is applied to the wave, to the beam and to the plate equations with boundary delays. Keywords se ond order evolution equations, wave equations, time-varying delay, stabilization, Lyapunov fun tional. 1 Introdu tion Time-delay often appears in many biologi al, ele tri al engineering systems and me hani al appli ations, and in many ases, delay is a sour e of instability [7?. In the ase of distributed parameter systems, even arbitrarily small delays in the feedba k may destabilize the system (see e.g. [5, 16, 24, 17?). The stability issue of systems with delay is, therefore, of theoreti al and pra ti al importan e. There are only a few works on Lyapunov-based te hnique for Partial Dif- ferential Equations (PDEs) with delay.
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