Solid Controllability in Fluid Dynamics

One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls

In this paper we study a mathematical model of one-dimensional swimmers performing a planar motion while fully immersed in a viscous fluid. The swimmers are assumed to be of small size, and all inertial effects are neglected. Hydrodynamic interactions are treated in a simplified way, using the local drag approximation of resistive force theory. We prove existence and uniqueness of the solution of the equations of motion driven by shape changes of the swimmer. Moreover, we prove a controllability result showing that given any pair of initial and final states, there exists a history of shape cha…

Micromotions and controllability of a swimming model in an incompressible fluid governed by 2D or 3D Navier--Stokes equations

We study the local controllability properties of 2D and 3D bio-mimetic swimmers employing the change of their geometric shape to propel themselves in an incompressible fluid described by Navier-Stokes equations. It is assumed that swimmers' bodies consist of finitely many parts, identified with the fluid they occupy, that are subsequently linked by the rotational and elastic internal forces. These forces are explicitly described and serve as the means to affect the geometric configuration of swimmers' bodies. Similar models were previously investigated in [6]-[13].

Controllability of a simplified fluid-structure interaction system

We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estima…

Remark on the global null controllability for a viscous Burgers-particle system with particle supported control

This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton's second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle. One of the novelties of our work is the fact that we achieve this controllability result in a uniform time for all initial data and without any smallness assumptions on the initial data.

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 implies mixing I. Convergence in the total variation metric

This paper is the first part of a project devoted to studying the interconnection between controllability properties of a dynamical system and the large-time asymptotics of trajectories for the associated stochastic system. It is proved that the approximate controllability to a given point and the solid controllability from the same point imply the uniqueness of a stationary measure and exponential mixing in the total variation metric. This result is then applied to random differential equations on a compact Riemannian manifold. In the second part, we shall replace the solid controllability by…

Lagrangian controllability at low Reynolds number

In this paper, we establish a result of Lagrangian controllability for a fluid at low Reynolds number, driven by the stationary Stokes equation. This amounts to the possibility of displacing a part of a fluid from one zone to another by suitably using a boundary control. This relies on a weak variant of the Runge-Walsh's theorem (on approximation of harmonic functions) concerning the Stokes equation. We give two variants of this result, one of which we believe to be better adapted to numerical simulations.

Approximate controllability of the linearized Boussinesq system in a two dimensional channel

In this paper, we prove an approximate controllability result for the linearized Boussinesq system around a fluid at rest, in a two dimensional channel, when the control acts only on the temperature, through the upper boundary.

一类非牛顿不可压缩粘性流体的局部零能控性

Local null controllability of a class of non-Newtonian incompressible viscous fluids

本文研究 Ladyzhenskaya 型非牛顿不可压缩粘性流体的局部零能控性，利用 Pontryagin 最小值原理和自举论证证明线性化 Stokes 问题光滑控制的存在性并推广到非线性情形，同时提出收敛的拟牛顿算法并通过数值实验验证。

完美流体中3D游泳者的通用可控性

Generic Controllability of 3D Swimmers in a Perfect Fluid

本文研究在完美流体中通过形状变化驱动的3D游泳者的跟踪控制问题，证明了通过微小叠加变形可实现任意轨迹跟踪（同步游泳），以及通过最多五种基本运动组合实现任意轨迹跟踪（自由泳）。

Controllability properties of a class of systems modeling swimming microscopic organisms

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploits the action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, with Stokes equations governing the surrounding fluid. The action of the cilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a system is generically controllable when the space of controlled velocity fields is at least three-dimensional. We also provide a complet…

通过控制不可压缩流体流动增强弛豫

Relaxation enhancement by controlling incompressible fluid flows

提出基于PDE可控性的方法增强被动标量场扩散混合，通过不可压缩Euler方程的状态轨迹构建时间依赖的弛豫增强向量场，并利用近似和精确可控性定理实现弛豫增强。

Navier-Stokes Equation on the Rectangle

We study controllability issues for the Navier-Stokes Equation on a two dimensional rectangle with so-called Lions boundary conditions. Rewriting the Equation using a basis of harmonic functions we arrive to an infinite-dimensional system of ODEs. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low modes forcing of finite-dimensional Galerkin approximations of that system. Proving the continuity of the ``control $\mapsto$ solution'' map in the so-called relaxation metric we use it to prove both solid controllability on observed component and $\ma…

Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system

In this second part we prove that the full nonlinear fluid-solid system introduced in Part I is stabilizable by deformations of the solid that have to satisfy nonlinear constraints. Some of these constraints are physical and guarantee the self-propelled nature of the solid. The proof is based on the boundary feedback stabilization of the linearized system. From this boundary feedback operator we construct a deformation of the solid which satisfies the aforementioned constraints and which stabilizes the nonlinear system. The proof is made by a fixed point method.

Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system

This paper is the first part of a work which consists in proving the stabilization to zero of a fluid-solid system, in dimension 2 and 3. The considered system couples a deformable solid and a viscous incompressible fluid which satisfies the incompressible Navier-Stokes equations. By deforming itself, the solid can interact with the environing fluid and then move itself. The control function represents nothing else than the deformation of the solid in its own frame of reference. We there prove that the velocities of the linearized system are stabilizable to zero with an arbitrary exponential d…

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…

The $N$-link swimmer in three dimensions: controllability and optimality results

The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by op…

A monolithic one-velocity-field optimal control formulation for fluid-structure interaction problems with large solid deformation

In this article, we formulate a monolithic optimal control method for general time-dependent Fluid-Structure Interaction (FSI) systems with large solid deformation. We consider a displacement-tracking type of objective with a constraint of the solid velocity, tackle the time-dependent control problems by a piecewise-in-time control, cope with large solid displacement using a one-velocity fictitious domain method, and solve the fully-coupled FSI and the corresponding adjoint equations in a monolithic manner. We implement the proposed method in the open-source software package FreeFEM++ and asse…

一维流固耦合模型的逆问题

Inverse problems for one-dimensional fluid-solid interaction models

该研究针对带时变界面的一维Burgers方程流固耦合模型，探究由区间一端的Dirichlet与Neumann数据反演界面形状的逆问题，建立了唯一性与条件稳定性估计，证明采用了基于Carleman和插值不等式的侧向估计方法。

Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions

We study the finite-horizon optimal control problem with quadratic functionals for an established fluid-structure interaction model. The coupled PDE system under investigation comprises a parabolic (the fluid) and a hyperbolic (the solid) dynamics; the coupling occurs at the interface between the regions occupied by the fluid and the solid. We establish several trace regularity results for the fluid component of the system, which are then applied to show well-posedness of the Differential Riccati Equations arising in the optimization problem. This yields the feedback synthesis of the unique op…

Regularity of boundary traces for a fluid-solid interaction model

We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system -- of interest in its own right -- establishes, in addition,…

使用一个边界控制的一维正压与非正压线性化可压缩 Navier-Stokes 系统的零能控性

Null controllability of one-dimensional barotropic and non-barotropic linearized compressible Navier-Stokes system using one boundary control

本文证明一维线性化可压缩 Navier-Stokes 方程在正压与非正压情形下均可由单个边界控制实现大时间零能控，并给出正则性与时间尖锐性及不可近似能控结论。

耦合KdV系统可控性的边界效应

Boundary Effects on the Controllability of Coupled KdV Systems

本文研究有界区间上耦合KdV系统的精确边界可控性，通过线性化系统的对偶方法、Paley-Wiener谱分析及压缩映射定理，在最多四个控制输入下建立了完整系统的局部可控性。