In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In...

Let there be given a differential operator on ${ℝ}^n$ of the form $D={\sum }_{i,j=1}^n{a}_{ij}·x_j\partial /\partial x_i+\mu$, where $A=\left(a_{ij}\right)$ is a real matrix and μ is a complex number. We study the following question: To what extent the mapping $D:S^{\text{'}}\left({ℝ}^n\right)\to S^{\text{'}}\left({ℝ}^n\right)$ is surjective? We shall give some conditions on A and μ which assure the surjectivity of D.

This paper deals with the diffusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diffusive limit, the right discrete diffusion equation with the same value of the diffusion coefficient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization...

This paper deals with the diffusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diffusive limit, the right discrete diffusion equation with the same value of the diffusion coefficient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization...

Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient...

Nous démontrons l’unicité des solutions faibles pour une classe d’équations de transport dont les vitesses sont partiellement à variations bornées. Nous nous intéressons à des champs de vecteurs du type$$a_1\left(x_1\right)·{\partial }_{x_1}+a_2(x_1,x_2)·{\partial }_{x_2},a_1\in BV\left({ℝ}_{x_1}^{N_1}\right),a_2\in L_{x_1}^1\left(BV\left({ℝ}_{x_2}^{N_2}\right)\right),$$avec une borne sur la divergence de chacun des champs $a_1,a_2$. Ce modèle a été étudié récemment dans [LL] par C. Le Bris et P.-L. Lions avec une régularité $W^{1,1}$ ; nous montrons ici également que, dans le cas $W^{1,1}$, le contrôle $L^{\infty }$ de la divergence totale du champ est suffisant. Notre méthode consiste à démontrer...

Whenever nonlinear problems have to be solved through approximation methods by solving related linear problems a priori estimates are very useful. In the following this kind of estimates are presented for a variety of equations related to generalized first order Beltrami systems in the plane and for second order elliptic equations in ${ℝ}^m$. Different types of boundary value problems are considered. For Beltrami systems these are the Riemann-Hilbert, the Riemann and the Poincaré problem, while for elliptic...