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

Stabilization of 2D Navier-Stokes equations by means of actuators with locally supported vorticity

Exponential stabilization to time-dependent trajectories for the incompressible Navier-Stokes equations is achieved with explicit feedback controls. The fluid is contained in two-dimensional spatial domains and the control force is, at each time instant, a linear combination of a finite number of given actuators. Each actuator has its vorticity supported in a small subdomain. The velocity field is subject to Lions boundary conditions. Simulations are presented showing the stabilizing performance of the proposed feedback. The results also apply to a class of observer design problems.Comment: 9 figure

Robust Output Regulation of the Linearized Boussinesq Equations with Boundary Control and Observation

We study temperature and velocity output tracking problem for a two-dimensional room model with the fluid dynamics governed by the linearized translated Boussinesq equations. Additionally, the room model includes finite-dimensional models for actuation and sensing dynamics, thus the complete model dynamics are governed by an ODE-PDE-ODE system. As the main result, we design a low-dimensional internal model based controller for robust output racking of the room model. Efficiency of the controller is demonstrated through a numerical example of velocity and temperature tracking.Comment: 26 pages, 9 figures, submitte

Robust Output Tracking for a Room Temperature Model with Distributed Control and Observation

We consider robust output regulation of a partial differential equation model describing temperature evolution in a room. More precisely, we examine a two-dimensional room model with the velocity field and temperature evolution governed by the incompressible steady state Navier-Stokes and advection-diffusion equations, respectively, which coupled together form a simplification of the Boussinesq equations. We assume that the control and observation operators of our system are distributed, whereas the disturbance acts on a part of the boundary of the system. We solve the robust output regulation problem using a finite-dimensional low-order controller, which is constructed using model reduction on a finite element approximation of the model. Through numerical simulations, we compare performance of the reduced-order controller to that of the controller without model reduction as well as to performance of a low-gain robust controller.Comment: 12 pages, 5 figures. Accepted for publication in the Proceedings of the 24th International Symposium on Mathematical Theory of Networks and Systems, 23-27 August, 202

Several questions concerning the control of parabolic systems

This paper is devoted to recall several recent results concerning the null controllability of some parabolic systems. Among others, we will consider the classical heat equation, the Burgers, Navier-Stokes and GinzburgLandau equations, etc.Dirección General de Investigación Científica y Técnic