Abstract. We p r e s e n t here a return method to describe some attainable sets on an interval of the classical Burger equation by means of the variation of the domain. Statement of the main resultWe are here interested in the following problem of controllability: Let T c be an arbitrary real number, X a given normed space of real functions of the real variable de ned on 1 2], z 1 z 0 elements of X. Does there exist a weak (entropic) solution of: where " > 0 is a priori given, and if this holds for any ( z 0 z 1 ) 2 X X and any " > 0, one says that there is approximate controllability i n X.According to the settlement of the controllability problem, we take here X = BV ( 1 2 here Df is the derivative o f f in the sense of measures.The following conditions (ES) are the conditions necessarily satis ed by spatial values of weak entropic solutions of the Cauchy problem (1.2) with initial value in the class of functions with bounded variation: we s a y that a Universit e d e V ersailles, Analyse appliqu ee,

In this paper, we consider the two-dimensional Euler equation in a bounded domain Ω, with a boundary control located on an arbitrary part of the boundary. We prove that, given two Jordan curves which are homotopic in Ω and which surround the same area, given an initial data and a positive time T , one can find a control such that the corresponding solution drives the first curve inside Ω arbitrarily close to the second one (in any C k norm) at time T. We also prove that given two vortex patches satisfying the same conditions on their contour, one can approximately deform the first one into the second one. Résumé. Dans cet article, nous considérons l'équation d'Euler des fluides parfaits incompressibles dans un domaine borné bidimensionnel, avec un contrôle frontière localisé sur une partie arbitraire du bord. Nous montrons qu'étant donnés deux courbes de Jordan homotopes et encerclant la même aire, une donnée initiale et un temps T strictement positif, on peut trouver un contrôle tel que la solution correspondante de l'équation d'Euler mène la première courbe vers la seconde de manière arbitrairement proche (en toute norme C k) au temps T. Nous montronségalement que sous la même condition sur les contours, on peut déformer de manière approchée une poche de tourbillon sur une autre dans le domaine.

In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain Ω. The matrixvalued coefficients A of such systems is our control in Ω and will be taken in L 2 (Ω; R N ×N ) which in particular may comprises the case of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may exhibit non-uniqueness of weak solutions-namely, approximable solutions as well as another type of weak solutions that can not be obtained through the L ∞ -approximation of matrix A. Following the direct method in the calculus of variations, we show that the given OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can not be attainable through the L ∞ -approximation of the original problem.

Abstract. We establish a result concerning the so-called Lagrangian controllability of the Euler equation for incompressible perfect fluids in dimension 3. More precisely we consider a connected bounded domain of R 3 and two smooth contractible sets of fluid particles, surrounding the same volume. We prove that given any initial velocity field, one can find a boundary control and a time interval such that the corresponding solution of the Euler equation makes the first of the two sets approximately reach the second one.

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