This paper deals with the problem of stabilizing a system in the presence ofsmall measurement errors. It is known that, for general stabilizable systems,there may be no possible memoryless state feedback which is robust withrespect to such errors. In contrast, a precise result is given here, showingthat, if a (continuous-time, finite-dimensional) system is stabilizable in anyway whatsoever (even by means of a dynamic, time varying, discontinuous,feedback) then it can also be semiglobally and practicallystabilized in a way which is insensitive tosmall measurement errors, by means of a hybrid strategy based on the idea ofsampling at a “slow enough” rate.

This article is devoted to the study of a perturbation with a viscosity termin an elliptic equation involving the p-Laplacian operator and related tothe best contant problem in Sobolev inequalities in the critical case.We prove first that this problem, together with the equation, is stableunder this perturbation, assuming some conditions on the datas. In thenext section, we show that the zero solution is strongly isolated in somesense, among the space of the solutions. Actually, we end the paper bygiving some analoguous results in the case where the datas presentsymmetries.

We give sufficient conditions which allow the study of the exponentialstability of systems closely related to the linear thermoelasticity systemsby a decoupling technique. Our approach is based on the multiplierstechnique and our result generalizes (from the exponential stability pointof view) the earlier one obtained by Henry et al.

We consider optimal distributed and boundary control problemsfor semilinear parabolic equations, where pointwise constraints onthe control and pointwise mixed control-state constraints of bottlenecktype are given. Our main result states the existence of regularLagrange multipliers for the state-constraints. Under naturalassumptions, we are able to show the existence of bounded and measurableLagrange multipliers. The method is based on results from the theoryof continuous linear programming problems.

Fast filtering algorithms arising from linear filtering and estimation are nonlineardynamical systems whose initial values are the statistics of the observation process. In this paper, we give a fairly complete description of the phase portrait for such nonlinear dynamical systems, as well as a special type of naturally related matrix Riccati equation.

On donne un développement asymptotique du profiliso pé ri mé tri que de ${\mathbb R}^n$ muni d'une métrique riemanniennepériodique, et des conséquences pour le problème de la formed'équilibre des cristaux.

Let Ω be a smooth bounded domain in ${\bf R}^n$ ,n > 1, let a and f be continuous functions on $\bar\Omega$ , $1^\star ={n\over n-1}$ . We are concerned here with the existence of solution in $BV(\Omega)$ , positive or not, to the problem:


$$\left\{\begin{array}{rl}-{\rm div}\ \sigma+a(x) sign\u &= f|u|^{1^\star-2} u\cr\sigma.\nabla u &= |\nabla u|\ {\rm in}\ \Omega\cru\ {\rm is\ not \ identically\ zero}, &-\sigma.n (u) = |u|\ {\rm on }\ \partial\Omega.\end{array}\right.$$

This problem is closely related to the extremal functions for the problem ofthe best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$ .