DCDS-S

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.

NHM

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.

MCRF

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.

DCDS

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).

DCDS-S

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.

NHM

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.

MCRF

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.

DCDS-S

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.

DCDS

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.

EECT

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.