Remarks on global controllability for the shallow-water system with two control forces

In this paper we deal with the compressible Navier-Stokes equations with a friction term in one dimension on an interval. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. Inspired by the work of Guerrero and Imanuvilov in \cite{GI} on the viscous Burger equation, we prove by choosing irrotational data and using the notion of effective velocity developed in \cite{cpde,cras} that the exact global controllability result does not hold for any time $T$

Global exponential stabilisation for the Burgers equation with localised control

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any $\varepsilon>0$, one can find a time $T$ of order $\log\varepsilon^{-1}$ such that any initial state can be steered to the $\varepsilon$-neighbourhood of a given trajectory at time $T$. This property combined with an earlier result on local exact controllability shows that the Burgers equation is globally exactly controllable to trajectories in a finite time. We also prove that the approximate controllability to arbitrary targets does not hold even if we allow infinite time of control.Comment: 19 page

On the uniform controllability for a family of non-viscous and viscous Burgers- α systems

In this paper we study the global controllability of families of the so called non-viscous and viscous Burgers- systems by using boundary and space independent distributed controls. In these equations, the usual convective velocity of the Burgers equation is replaced by a regularized velocity, induced by a Helmholtz lter of characteristic wavelength . First, we prove a global exact controllability result (uniform with respect to ) for the non-viscous Burgers- system, using the return method and a xed-point argument. Then, the global uniform exact controllability to constant states is deduced for the viscous equations. To this purpose, we rst prove a local exact controllability property and, then, we establish a global approximate controllability result for smooth initial and target states

Uniform local null control of the Leray-α model

This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0+ to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.Instituto Nacional de Ciência e Tecnologia de MatemáticaCoordenação de aperfeiçoamento de pessoal de nivel superiorConselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Educación y Ciencia (España) MTM2006-07932 MTM2010-1559

Controllability and observabiliy of an artificial advection-diffusion problem

In this paper we study the controllability of an artificial advection-diffusion system through the boundary. Suitable Carleman estimates give us the observability on the adjoint system in the one dimensional case. We also study some basic properties of our problem such as backward uniqueness and we get an intuitive result on the control cost for vanishing viscosity.Comment: 20 pages, accepted for publication in MCSS. DOI: 10.1007/s00498-012-0076-

Simultaneous local exact controllability of 1D bilinear Schr\"odinger equations

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schr\"odinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay

Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes , etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants

Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

This paper deals with the null-controllability of a system of mixed parabolic-elliptic pdes at any given time $T>0$. More precisely, we consider the Kuramoto-Sivashinsky--Korteweg-de Vries equation coupled with a second order elliptic equation posed in the interval $(0,1)$. We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the Carleman approach we provide the existence of a control with the explicit cost $Ke^{K/T}$ with some constant $K>0$ independent in $T$. Then, applying the source term method and the Banach fixed point argument, we conclude the small-time local null-controllability result of the nonlinear systems. Besides, we also established a uniform null-controllability result for an asymptotic two-parabolic system (fourth and second order) that converges to the concerned parabolic-elliptic model when the control is acting on the second order pde