Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback
Serge Nicaise Cristina Pignotti
Discrete & Continuous Dynamical Systems - S 2016, 9(3): 791-813 doi: 10.3934/dcdss.2016029
We study the stabilization problem for the wave equation with localized Kelvin--Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the result of [19] where, in a more general setting, the case of distributed structural damping is considered.
keywords: delay feedbacks Wave equation stabilization.
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks
Serge Nicaise Julie Valein
Networks & Heterogeneous Media 2007, 2(3): 425-479 doi: 10.3934/nhm.2007.2.425
In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.
keywords: wave equation stabilization delay
Control and stabilization of 2 × 2 hyperbolic systems on graphs
Serge Nicaise
Mathematical Control & Related Fields 2017, 7(1): 53-72 doi: 10.3934/mcrf.2017004

We consider 2× 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at all except one external vertices. This result is obtained with the help of a perturbation argument and an observability estimate for an associated wave type equation. An exact controllability result is also deduced.

keywords: Hyperbolic systems wave equation stabilization
Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation
Denis Mercier Serge Nicaise
Discrete & Continuous Dynamical Systems - A 1998, 4(2): 273-300 doi: 10.3934/dcds.1998.4.273
In this paper, we first prove existence results for general systems of differential equations of parabolic and hyperbolic type in a Hilbert space setting using the notion of Agmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessary and sufficient condition on the boundary and/or transmission conditions which insures the dissipativity of the (spatial) operators. Our second goal is to take advantage of the one-dimensional structure of networks in order to build appropriate prewavelet bases in view to the numerical approximation of the above problems. Indeed we show that the use of such bases for their approximation (by the Galerkin method for elliptic operators and a fully discrete scheme for parabolic ones) leads to linear systems which can be preconditioned by a diagonal matrix and then can be reduced to systems with a condition number uniformly bounded (with respect to the mesh parameter).
keywords: systems Networks differential equations prewavelets.
Optimal control of some quasilinear Maxwell equations of parabolic type
Serge Nicaise Fredi Tröltzsch
Discrete & Continuous Dynamical Systems - S 2017, 10(6): 1375-1391 doi: 10.3934/dcdss.2017073

An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak Gâteaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and an associated adjoint calculus are derived.

keywords: Evolution Maxwell equations parabolic equation quasilinear equation vector potential formulation optimal control necessary optimality conditions adjoint equation
Asymptotic analysis of a simple model of fluid-structure interaction
Serge Nicaise Cristina Pignotti
Networks & Heterogeneous Media 2008, 3(4): 787-813 doi: 10.3934/nhm.2008.3.787
This paper is devoted to the asymptotic analysis of simple models of fluid-structure interaction, namely a system between the heat and wave equations coupled via some transmission conditions at the interface. The heat part induces the dissipation of the full system. Here we are interested in the behavior of the model when the thickness of the heat part and/or the heat diffusion coefficient go to zero or to infinity. The limit problem is a wave equation with a boundary condition at the interface, this boundary condition being different according to the limit of the above mentioned parameters. It turns out that some limit problems are dissipative but some of them are non dissipative or their behavior is unknown.
keywords: Fluid-structure system asympotic behavior
Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks
Serge Nicaise
Mathematical Control & Related Fields 2011, 1(3): 331-352 doi: 10.3934/mcrf.2011.1.331
A Mindlin-Timoshenko model with non constant and non smooth coefficients set in a bounded domain of $\mathbb{R}^d, d\geq 1$ with some internal dissipations is proposed. It corresponds to the coupling between the wave equation and the dynamical elastic system. If the dissipation acts on both equations, we show an exponential decay rate. On the contrary if the dissipation is only active on the elasticity equation, a polynomial decay is shown; a similar result is proved in one dimension if the dissipation is only active on the wave equation.
keywords: Mindlin-Timoshenko system stabilization.
Stability of the heat and of the wave equations with boundary time-varying delays
Serge Nicaise Julie Valein Emilia Fridman
Discrete & Continuous Dynamical Systems - S 2009, 2(3): 559-581 doi: 10.3934/dcdss.2009.2.559
Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be time-varying with an a priori given upper bound on its derivative, which is less than $1$. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.
keywords: stability Heat equation wave equation time-varying delay Lyapunov functional.
Polynomial stabilization of some dissipative hyperbolic systems
Kais Ammari Eduard Feireisl Serge Nicaise
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4371-4388 doi: 10.3934/dcds.2014.34.4371
We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
keywords: observability inequality polynomial stability dissipative hyberbolic system acoustic equation. Exponential stability resolvent estimate
Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems
Farah Abdallah Denis Mercier Serge Nicaise
Evolution Equations & Control Theory 2013, 2(1): 1-33 doi: 10.3934/eect.2013.2.1
In this paper, we consider two damped wave problems for which the damping terms are allowed to change their sign. Using a careful spectral analysis, we find critical values of the damping coefficients for which the problem becomes exponentially or polynomially stable up to these critical values.
keywords: Indefinite damping spectral analysis. Riesz basis

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