On the Fattorini Criterion for Approximate Controllability and Stabilizability of Parabolic Systems

In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type $y'=A y+Bu$. We precise the result proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, in the case where $B$ can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini's criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier-Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini's criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems

Uniform stabilization of 3D Navier-Stokes equations in critical Besov spaces with finite dimensional, tangential-like boundary, localized feedback controllers

The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of a `minimal' and `least' invasive feedback strategy which consists of a control pair $\{ v,u \}$ \cite{LT2:2015}. Here $v$ is a tangential boundary feedback control, acting on an arbitrary small part $\widetilde{\Gamma}$ of the boundary $\Gamma$; while $u$ is a localized, interior feedback control, acting tangentially on an arbitrarily small subset $\omega$ of the interior supported by $\widetilde{\Gamma}$. The ideal strategy of taking $u = 0$ on $\omega$ is not sufficient. A question left open in the literature was: Can such feedback control $v$ of the pair $\{ v, u \}$ be asserted to be finite dimensional also in the dimension $d = 3$? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control $v$, it is here then necessary to abandon the Hilbert setting of past literature and replace it with a Besov setting which are `close' to $L^3(\Omega)$ for $d=3$. It is in line with recent critical well-posedness in the full space of the non-controlled N-S equations. A key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions. The proof is constructive and is "optimal" also regarding the "minimal" number of tangential boundary controllers needed. The new setting requires establishing maximal regularity in the required critical Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. Finally, the minimal amount of tangential boundary action is linked to the issue of unique continuation of over-determined Oseen eigenproblems

Uniform Stabilization of Navier-Stokes Equations in Lq-based Sobolev and Besov Spaces

We consider 2- or 3-dimensional Navier-Stokes equations defined on a bounded domain $\Omega$ subject to an external force, assumed to cause instability. We then seek to uniformly stabilize such N-S system, in the vicinity of an unstable equilibrium solution, in $L^q$-based Sobolev and Besov spaces, by finite dimensional feedback controls. This work is divided in to two parts. In Part I, the finite dimensional feedback controls are localized on an arbitrarily small open interior subdomain $\omega$ of $\Omega$. Instead, in Part II seeks tangential boundary feedback stabilizing controls. It provides a solution to the following recognized open problem in the theory of uniform stabilization of d-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of tangential boundary localized feedback controls: can these stabilizing controls be asserted to be finite dimensional also in the physical dimension $d=3$? To achieve the desired finite dimensionality result of the feedback tangential boundary controls, it was then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with an appropriate Lq-based/Besov setting with tight parameters related to the physical dimension $d$, where the compatibility conditions are not recognized. This result is also a new contribution to the area of maximal regularity with inhomogeneous boundary feedback

Internal exponential stabilization to a nonstationary solution for 1D Burgers equations with piecewise constant controls

International audienceThe feedback stabilization of the Burgers system to a nonstationary solution using a finite number of internal piecewise constant controls is considered. Estimates for the number of needed controls are derived. In the particular case of no constraint on the support of the control a better estimate is derived, so the possibility of getting an analogous estimate for the general case is discussed.That possibility is suggested by the results of some numerical simulations

Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations

International audienceThe feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the controla better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examplesare presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimatein the general case analogous to that in the particular one is plausible

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

Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. Application to the Navier-Stokes system

International audienceLet $A : \mathcal{D}(A)\to \mathcal{X}$ be the generator of an analytic semigroup and $B : \mathcal{U} \to [{\cal D}(A^*)]'$ a quasi-bounded operator. In this paper, we consider the stabilization of the system $y'=Ay+Bu$ where $u$ is the linear combination of a family $(v_1,\ldots,v_K)$. Our main result shows that if $(A^*,B^*)$ satisfies a unique continuation property and if $K$ is greater or equal to the maximum of the geometric multiplicities of the the unstable modes of $A$, then the system is generically stabilizable with respect to the family $(v_1,\ldots,v_K)$. With the same functional framework, we also prove the stabilizability of a class of nonlinear system when using feedback or dynamical controllers. We apply these results to stabilize the Navier--Stokes equations in 2D and in 3D by using boundary control