An Example on the Effect of Time Delays in Boundary Feedback Stabilization of Wave Equations

Feedback stabilization of a class of evolution equations with delay

“…These results have been obtained in the case of one dimensional systems in [29] by a direct spectral method. The robustness of the time delay in the boundary or the internal stabilization of the wave and other equation, has been studied in [7,8,[14][15][16].…”

Existence and asymptotic stability of a viscoelastic wave equation with a delay

“…• It would be interesting to investigate the viscoelastic wave equation with a time delay in the boundary condition and a velocity term in the equation. Concerning the wave equation, Datko et al [9] treated the one-dimensional problem ⎧ ⎨ ⎩ u tt (x, t) − u xx (x, t) + 2au t (x, t) + a 2 u(x, t) = 0, 0 < x < 1, t > 0, u(0, t) = 0,…”

“…In [9], the authors showed that a small delay in a boundary control is a source of instability. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary; see for instance [17,18,25] and references therein.…”

“…Using the potential well theory, Tatar and Messaoudi [15] extended the result in [5] to a situation where a source term of the form |u| p−2 u is present in Eq. (9). Recently, Messaoudi and Tatar [16] have studied (9) with (γ = 0) and showed that the viscoelastic damping term is strong enough to stabilize the system.…”

On the Feedback Control of the Wave Equation

“…The feedback controller is shown to consist of a velocity sensor, a proportional gain and a time delay. Datko et al [4] investigate the e!ects of time delays in boundary feedback stabilization schemes for wave equations.…”