Lagrangian controllability at low Reynolds number

Enhanced controllability of low Reynolds number swimmers in the presence of a wall

Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers which is the regime of interest for micro-organisms and micro-robots. We focus on self-propelled stokesian robots composed of assemblies of balls and we prove that the presence of a wall has an effect on their motility. More precisely, we demonstrate that a controllable swimmer remains controllable in a half space whereas the reachable set of a non fully controllable one is affected by the presence of a wall.

Lagrangian controllability of inviscid perfect fluids: a constructive approach

We present here a constructive method of Lagrangian approximate control- lability for the Euler equation. We emphasize on different options that could be used for numerical recipes: either, in the case of a bi-dimensionnal fluid, the use of formal com- putations in the framework of explicit Runge approximations of holomorphic functions by rational functions, or an approach based on the study of the range of an operator by showing a density result. For this last insight in view of numerical simulations in progress, we analyse through a simplified problem the observed instabilities.

Controllability of 3D Low Reynolds Swimmers

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequ…

On the controllability of the non-isentropic 1-D Euler equation

In this paper, we examine the question of the boundary controllability of the one-dimensional non-isentropic Euler equation for compressible polytropic gas, in the context of weak entropy solutions. We consider the system in Eulerian coordinates and the one in Lagrangian coordinates. We obtain for both systems a result of controllability toward constant states (with the limitation $γ< 5/3$ on the adiabatic constant for the Lagrangian system). The solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation.

Lagrangian Controllability of the 1-D Korteweg-de Vries Equation

We consider in this paper the problem of the Lagrangian controllability for the Korteweg-de Vries equation. Using the $N$-solitons solution, we prove that, for any length of the spatial domain $L>0$ and any time $T>0$, it is possible to choose appropriate boundary controls of KdV equation such that the flow associated to this solution exit the domain in time $T$.

使用拉格朗日方法的抛物型偏微分方程数值零能控性

Numerical null controllability of parabolic PDEs using Lagrangian methods

本文提出基于拉格朗日与增广拉格朗日方法并结合无界时间权重的数值框架，求解多维抛物型方程在小子集分布控制下的零能控性问题，并通过2D和3D数值实验验证其有效性。