We analyze the controllability of the motion of a fluid by means of the action of a vibrating shell coupled at the boundary of the fluid. The model considered is linear. We study its approximate controllability, i.e. whether the fluid may reach a dense set of final configurations at a given time. We show that this problem can be reduced to a unique continuation question for the Stokes system. We prove that this unique continuation property holds generically among analytic domains and therefore, that there is approximate controllability generically. We also prove that this result fails when Ω is a ball showing that the analyticity assumption on the domain is not sufficient.

An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be effective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry.We develop two computational methods based on this idea.One method results in a nonlinear time-dependant partial differential equation for the level-set function whose evolution minimizes the residual in the data fit. The second method is an optimization that generates a sequence of level-set functions that reduces the residual. The methods are illustrated in two applications : a deconvolution problem and a diffraction screen reconstruction problem.

For boundary or distributed controls, we get an approximate controllability result for the Navier-Stokes equations in dimension 2 in the case where the fluid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.

For a given U(1)-bundle E over M = $\lambda _{2}$ \{x1, ..., xn }, where the xi are n distinct points of $\lambda _{2}$ , we minimise the U(1)-Higgs action and we make an asymptotic analysis of the minimizers when the coupling constant tends to infinity. We prove that the curvature (= magnetic field) converges to a limiting curvature that we give explicitely and which is singular along line vortices which connect the xi . This work is the three dimensional equivalent of previous works in dimension two (see [3] and [4]). The results presented here were announced in [12].

Lower semicontinuity of multiple integrals ∫Ωƒ(x,uα,Pα)dµ and ∫Ωƒα(x,uα,Pα)dµ are studied. It is proved that the two can derive from each other under certain general hypotheses such as uniform lower compactness property and locally uniform convergence of ƒα to ƒ. The result is applied to obtain some general lower semicontinuity theorems on multiple integrals with quasiconvex integrand ƒ, while ƒα are not required to be quasiconvex.

We show how the use of a parallel between the ordinary (+, X) and the (max, +) algebras, Maslov measures that exploit this parallel, and more specifically their specialization to probabilities andthe corresponding cost measures of Quadrat, offer a completely parallel treatment of stochastic and minimax control of disturbed nonlinear discrete time systems with partial information. This paper is based upon, and improves, the discrete time part of the earlier paper [9].

For continuous time semilinear control systems with constrained control values stabilizing discrete feedback controls are discussed. It is shown that under an accessibility condition exponentialdiscrete feedback stabilizability is equivalent to open loop exponential asymptotic null controllability. A numerical algorithm for the computation of discrete feedback controls is presented and anumerical example is discussed.

Multi-level saturation feedbacks induce nonlinear disturbance-to-state L 2 stability for nonlinear systems in feedforward form. This class of systems includes linear systems with actuatorconstraints.

In this paper we study the problem of the car with N trailers. It was proved in previous works ([9], [12]) that when each trailer is perpendicular with the previous one the degree of nonholonomy isFn+3 (the (n+3)-th term of the Fibonacci's sequence) and that when no two consecutive trailers are perpendicular this degree is n+2. We compute here by induction the degree of non holonomyin every state and obtain a partition of the singular set by this degree of non-holonomy. We give also for each area a set of vector fields in the Lie Algebra of the control system wich makes a basis of the tangent space.

The goal of this article is the study of the approximate controllability for two approximations of Navier Stokes equations with distributed controls. The method of proof combines a suitable linearization of the system with a fixed point argument. We then are led to study the approximate controllability of linear Stokes systems with potentials. We study both the case where there is no constraint on the control and the case where we search a control having one null component. In both cases, the problems is reduced to prove unique continuation results. This is done by means of Carleman estimates.