Sergio Guerrero, Pierre Cornilleau
In this paper, we study the local exact controllability to special trajectories of the micropolar fluid systems in dimension d = 2 and d = 3. We show that controllability is possible acting only on one velocity.
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In this paper we study the controllability of a coupled Keller-Segel-Navier-Stokes system. We show the local exact controllability of the system around some particular trajectories. The proof relies on new Carleman inequalities for the chemotaxis part and some improved Carleman inequalities for the Stokes system.
Boundary controllability of incompressible Euler fluids with Boussinesq heat effects
本文研究热效应重要的无粘不可压缩流体的边界可控性，通过Boussinesq近似建模，在零热扩散情形下建立了二维和三维流动中速度场和温度的同步全局精确可控性，并在正热扩散系数下给出了速度场精确可控性和温度局部零可控性的附加结果。
We prove the exact controllability result to trajectories of a simplified model of motion of a rigid body in fluid flow. Unlike a previously know results such a trajectory does not need to be a stationary solution.
We survey results of recent activity towards studying controllability and accessibility issues for equations of dynamics of incompressible fluids controlled by low-dimensional or, degenerate, forcing. New results concerning controllability of Navier-Stokes/Euler systems on two-dimensional sphere and on a generic two-dimensional domain are represented.
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.
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].
We study an optimal boundary control problem for the two-dimensional stationary micropolar fluids system with variable density. We control the system by considering boundary controls, for the velocity vector and angular velocity of rotation of particles, on parts of the boundary of the flow domain. On the remaining part of the boundary, we consider mixed boundary conditions for the vector velocity (Dirichlet and Navier conditions) and Dirichlet boundary conditions for the angular velocity. We analyze the existence of a weak solution obtaining the fluid density as a scalar function of the strea…
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…
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.
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.
In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around $(\bar ρ,0,\bar θ)$, with $\bar ρ> 0,$ $ \bar θ> 0$ is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove tha…
Controllability and kinetic limit of spherical particles immersed in a viscous fluid
本文研究粘性流体中球形颗粒系统的可控性与动力学极限，分别利用几何控制理论和Boltzmann型表述分析少颗粒可控性及颗粒数趋于无穷时的极限行为。
Complex industrial processes such as the drying of combustible biomass can be modeled with computational fluid dynamics simulations. Due to their complexity, it is not straightforward to use these models for the analysis of system properties or for solving optimal control problems. We show reduced order models can be derived and used for these purposes for industrial drying processes.
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.
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。
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…
The paper is devoted to the controllability problem for 3D compressible Euler system. The control is a finite-dimensional external force acting only on the velocity equation. We show that the velocity and density of the fluid are simultaneously controllable. In particular, the system is approximately controllable and exactly controllable in projections
In this work, we investigate the small-time global exact controllability of the Navier-Stokes equation, both towards the null equilibrium state and towards weak trajectories. We consider a viscous incompressible fluid evolving within a smooth bounded domain, either in 2D or in 3D. The controls are only located on a small part of the boundary, intersecting all its connected components. On the remaining parts of the boundary, the fluid obeys a Navier slip-with-friction boundary condition. Even though viscous boundary layers appear near these uncontrolled boundaries, we prove that small-time glob…
We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus $\mathbb{T}^2$. We assume the system to be controlled by a degenerate forcing applied to fixed number of modes. In our previous work \cite{ASpb,AS43,ASDAN} we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity ($ν>0$). Methods of differential geometric/Lie algebraic control theory have been used for tha…
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…
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）。
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。
Global controllability of Boussinesq channel flows only through the temperature
该研究针对平面周期槽道中带粘性扩散的Boussinesq系统，提出仅用薄条带温度控制实现全局近似可控的方法，结合小时间尺度论证与Coron返回法，取得该类系统非周期边界下首个全局可控性结果。
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…
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…
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。
Small-time global exact controllability to the trajectories for the viscous Boussinesq system
本文针对二维和三维光滑有界域内带特定边界条件的粘性Boussinesq系统，通过区域延拓、借鉴Navier-Stokes方程能控性策略及线性化加Carleman估计三步法，证明了边界小区域控制下的轨迹小时间全局精确能控性。
Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct
本文研究矩形管道中理想不可压缩MHD的全局精确可控性，利用return method和类压力校正子处理速度与磁场的非线性耦合，并通过初始数据延拓和加权空间等专门构造克服控制困难。
There are numerous ways to control objects in the Stokes regime, with microscale examples ranging from the use of optical tweezers to the application of external magnetic fields. In contrast, there are relatively few explorations of theoretical controllability, which investigate whether or not refined and precise control is indeed possible in a given system. In this work, seeking to highlight the utility and broad applicability of such rigorous analysis, we recount and illustrate key concepts of geometric control theory in the context of multiple particles in Stokesian fluids interacting with …
Controllability of Boussinesq flows driven by finite-dimensional and physically localized forces
本文证明二维 Boussinesq 流可由支撑于任意固定区域的有限维局域外力近似控制，并结合退化周期 Euler 流、Coron 返回方法与浮力机制构造了维数不依赖区域选择的低维控制空间。
On the control of some coupled systems of the Boussinesq kind with few controls
本文研究Boussinesq型耦合系统的局部精确轨迹可控性，通过减少控制数量，证明仅需对温度$θ$和浓度$c$方程进行空间局部控制即可实现目标。
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