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

Mehdi Badra (1) , Takéo Takahashi (2, 3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 CORIDA - Robust control of infinite dimensional systems and applications
3 EDP - Equations aux dérivées partielles

In this paper, we extend an approximate controllability criterion for infinite dimensional linear systems of type $y'=A y+Bu$, originally proved by H. O. Fattorini in \cite{Fattorini1966} for bounded input $B$, to the case where $B$ is unbounded. We also prove that if Fattorini criterion is satisfied and if the set of geometric multiplicities of $A$ is bounded then approximate controllability can be achieved with a finite dimensional control. Thus, we show that Fattorini criterion implies the feedback stabilizability of linear and nonlinear parabolic systems. When considering systems discribed by partial differential equations such a criterion reduces to a unique continuation theorem for an eigenvalue problem. We then consider flow systems described by coupled Navier-Stokes type equations (such as MHD system or micropolar fluid system) and we sketch a systematic procedure relying on Fattorini criterion for checking stabilizability of such nonlinear system. In particular, we provide local Carleman inequalities for Stokes equations that permit to prove unique continuation theorems related to the stabilizability of coupled Stokes type systems.

Mots clés: coupled; Stokes and MHD system; Approximate controllability; stabilizability; parabolic equation; finite dimensional control

Domaines: Equations aux dérivées partielles [math.AP]

Mehdi Badra, Takéo Takahashi. On the Fattorini Criterion for Approximate Controllability and Stabilizability of Parabolic Systems. 2012. ⟨hal-00743899v1⟩