We prove an estimate of Carleman type for the one dimensional heat equationwhere a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equationwhere (t, x) ∈ (0, T ) × (0, 1), ω = (α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.

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This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t -> +infinity. Our approach is based on integral inequalities and multiplier techniques. These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system. (C) 2007 Elsevier Inc. All rights reserved

The Liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations. Extensions to infinite dimensional settings have been studied by several authors (see e.g. Haraux, 1991 [11], and Komornik and Zuazua, 1990 [17] and references therein). Results on optimal energy decay rates under general conditions of the feedback is far from being complete. The purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation. We present a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for these systems, and on comparison principles based on the dissipation property, and interpolation inequalities (in the infinite dimensional case) for lower bounds of the energy. We stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi-discretized nonlinear damped vibrating PDE's. A part of this approach has been introduced in Alabau-Boussouira (2004, 2005) [1,2]. In the present paper, we identify a new, simple and explicit criteria to select a class of nonlinear feedbacks, for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in Alabau-Boussouira (2004 [1,2]. Moreover, we prove optimality of the decay rates for this class, in the finite dimensional case. This class includes a wide range of feedbacks, ranging from very weak nonlinear dissipation (exponentially decaying in a neighborhood of zero), to polynomial, or polynomial-logarithmic decaying feedbacks at the origin. In the infinite dimensional case, we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation. This principle relies on suitable interpolation inequalities. It allows us to give lower bounds for the energy of smooth initial data for the one-dimensional wave equation with a distributed polynomial damping, which improves Haraux (1995) [12] lower estimate of the energy for this case. We also establish lower bounds in the multi-dimensional case for sufficiently smooth solutions when such solutions exist. We further mention applications of these various results to several classes of PDE's, namely: the locally and boundary damped multi-dimensional wave equation, the locally damped plate equation and the globally damped coupled Timoshenko beams system but it applies to several other examples. Furthermore, we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems. We illustrate these results on the one-dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples. These optimal rates are not uniform with respect to the discretization parameter. We also discuss and explain why optimality results have to be stated diff...

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter µa > 0. We establish observability inequalities for weakly (when µa ∈ [0, 1[) as well as strongly (when µa ∈ [1, 2[) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when µa > 2) and the blow-up of the observability time when µa converges to 2 from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when µa ∈ [0, 2[. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary damped wave equation, for an arbitrarily growing nonlinear feedback close to the origin. This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of [1,2] together with the results of the previous section, to perform this stability analysis.

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Abstract. In this paper we investigate the exact controllability of n × n first order quasilinear hyperbolic systems by m < n internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due M. Gromov. However, in contrast with this prior article where a standard fixed point argument could be used to treat the nonlinearities, we use here a fixed point theorem of Nash-Moser type due to M. Gromov in order to handle the problem of loss of derivatives.Key words. Quasilinear hyperbolic systems, exact internal controllability, controllability of systems, algebraic solvability.AMS subject classifications. 35L50, 93B05, 93C10.1. Introduction. In this paper we investigate the exact controllability of n × n first order quasilinear hyperbolic systems by m < n internal controls that are localized in space in some part of the domain. While the controllability of quasilinear hyperbolic systems by boundary controls has been intensively studied, [Cir69, LR02, LR03, Wan06, Zha09, LRW10], to our knowledge there are no equivalent results for the internal controllability. On the other hand, the controllability of systems of PDEs with a reduced number of controls has been a challenging problem for the last decades, see for instance [AB03, ABL12, AB13, DLRL14 Navier-Stokes equations. Let us also point out that, in many of these articles, the general strategy is to start with a controllability result in the case where there are as many controls as the number of equations and then to try to remove some of these controls by a suitable procedure. We also follow this general strategy here.In [LR03], the authors introduced a constructive method to control quasilinear systems of n equations by n boundary controls. This proficient method is based on existence and uniqueness results of semi-global solutions [LJ01] (i.e. with large time and small data) that they apply to several mixed initial-boundary value problems, using also the equivalent roles of the time and the space. As we shall see below, using a method of extension of the domain (as it is often used in the parabolic framework), we can recover this result for the internal controllability, that is we can prove the controllability of n × n quasilinear systems by n internal controls. The situation is more complicated when we have less controls than ...