Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks

By Serge Nicaise, Cristina Pignotti. SIAM Journal of Control and Optimization, 2006

712 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/cd2c0f8efecbbc9f54698ce0d0b539589b7f9948

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Citations in the corpus

Feedback stabilization of a class of evolution equations with delay (2009, methodology): “Recently, Nicaise–Pignotti adopted in [25,26] the same method for the wave equation with boundary or interior delay.”
Existence and asymptotic stability of a viscoelastic wave equation with a delay (2011, methodology): “• As we have mentioned above, for the wave equation, some instability results have been shown in [ 17 ] in the case μ2 ≥ μ1. It would be interesting to study the case μ2 >μ 1 in our problem (1).”
STABILITY OF THE HEAT AND OF THE WAVE EQUATIONS WITH BOUNDARY TIME-VARYING DELAYS (2009, neutral): “Indeed in [10, 12], the energy is decreasing under the assumption µ 1 µ 0 .”
Exponential stability of abstract evolution equations with time delay (2014, neutral): “[9]) the new variable Z(t, ρ) := BU(t− τρ), ρ ∈ (0, 1), t > 0 .”
General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback (2012, background): “In [21], Nicaise and Pignotti examined (1.”
Stabilization of the wave equation with boundary or internal distributed delay (2008, background): “As proved in [20] in this case there are instability phenomena for some delays τ.”
A Blow-Up Result in a Nonlinear Wave Equation with Delay (2016, background): “In the presence of delay (a > 0), Nicaise and Pignotti [20] examined system (1.”
Exponential stability of the wave equation with boundary time-varying delay (2010, methodology): “We are interested in the eect of a time{varying delay in boundary stabilization of the wave equation in domains of IRn: Delay eects arise in many pratical problems and it is well known that they can induce some unstabilities, see [5, 6, 7, 25 , 30].”
STABILIZATION OF SECOND ORDER EVOLUTION EQUATIONS WITH UNBOUNDED FEEDBACK WITH DELAY (2010, methodology): “2) is well-posed, where we closely follow the approach developed in [16] for the wave equation.”
Stability analysis of Euler‐Bernoulli beam with input delay in the boundary control (2012, background): “in [16] and [17] extended this result to a network of 1-d wave equations and n-dimensional wave equations.”
ASYMPTOTIC BEHAVIOR FOR A VISCOELASTIC WAVE EQUATION WITH A DELAY TERM (2013, background): “Many authors have focused on this problem and several results concerning existence, decay and instability have been obtained, see [5-7,10,19-24,26] and reference therein.”