Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force

In the Eulerian approach, the motion of an incompressible fluid is usually described by the velocity field which is given by the Navier--Stokes system. The velocity field generates a flow in the space of volume-preserving diffeomorphisms. The latter plays a central role in the Lagrangian description of a fluid, since it allows to identify the trajectories of individual particles. In this paper, we show that the velocity field of the fluid and the corresponding flow of diffeomorphisms can be simultaneously approximately controlled using a finite-dimensional external force. The proof is based on some methods from the geometric control theory introduced by Agrachev and Sarychev

Global approximate controllability for Schr\"odinger equation in higher Sobolev norms and applications

We prove that the Schr\"odinger equation is approximately controllable in Sobolev spaces $H^s$, $s>0$ generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schr\"odinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphere $S$ in $L^2$

Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations

International audienceWe consider a system of an arbitrary number of \textsc{1d} linear Schrödinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these $N$ equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument

Stochastic CGL equations without linear dispersion in any space dimension

We consider the stochastic CGL equation $$\dot u- \nu\Delta u+(i+a) |u|^2u =\eta(t,x),\;\;\; \text {dim} \,x=n,$$ where $\nu>0$ and $a\ge 0$, in a cube (or in a smooth bounded domain) with Dirichlet boundary condition. The force $\eta$ is white in time, regular in $x$ and non-degenerate. We study this equation in the space of continuous complex functions $u(x)$, and prove that for any $n$ it defines there a unique mixing Markov process. So for a large class of functionals $f(u(\cdot))$ and for any solution $u(t,x)$, the averaged observable \E f(u(t,\cdot)) converges to a quantity, independent from the initial data $u(0,x)$, and equal to the integral of $f(u)$ against the unique stationary measure of the equation

Global exact controllability in infinite time of Schr\"odinger equation: multidimensional case

We prove that the multidimensional Schr\"odinger equation is exactly controllable in infinite time near any point which is a finite linear combination of eigenfunctions of the Schr\"odinger operator. We prove that, generically with respect to the potential, the linearized system is controllable in infinite time. Applying the inverse mapping theorem, we prove the controllability of the nonlinear system