European Journal of Control (2001)7: 17-28 © 2001 EUCA European Journal of Control Remarks on the Feedback Stabilization of System Affine in Control L. Berrahmoune l , Y. Elboukfaoui 2 ,* and M. Erraoui 2 ,t I Departement de Mathematiques, ENS de Rabat BP 5118 Rabat, Morocco; 2Departement de Mathematiques, Faculte des Sciences Semialia BP 2390 Universite Cadi Ayyad, Marrakech, Morocco manifolds theorem have been used by many authors for the construction of the stabilizing feedback law (see for instance [2-5,23-25] and references therein). This paper is concerned with the stabilization of infi- nite dimensional systems described by the following abstract differential equation Here A is the infinitesimal generator of a Co-semigroup (etA)t>o on the separable complex Hilbert space H (norm 11·11, inner product ( . , . )), B is a locally Lipschitz mapping on Hand u(t) is a «:::-valued control. Recall that in the finite dimensional case (x(·) E jR", A and B are constant matrices of dimension (n x n), the well-known Jurdjevic-Quinn Theorem [19] (see also [22]) asserts that the system (1) is globally asymptoti- cally stabilizable at the origin by the feedback law In this paper we study the feedback stabilization problem for the control-affine system x = Ax + uBx. The operator A is assumed to be the generator of a compact Co-semigroup of contractions in a separable complex Hilbert space H; B is a locally Lipschitz mapping on H. First, based on the decomposition of contraction semigroup, we present sufficient conditions for stabilization of the system by means of thefeedback law u =- (x, Bx). Second, we show that under general assumptions all solutions of the closed loop system are asymptotic to the set of equilibria E as t tends to infinity. This result offers sufficient conditions for global stabilization of the system at the origin. Finally, we consider the question of robustness of the above feedback law. In other words we show that the stability property of the system, with the same control law u =- (x, Bx), remains invariant under certain classes of perturbations of the generator A. AMS Subject (1990): 93D IS, 93D22 Keywords: Stabilizability; Canonical decomposition of semigroup; Robustness; System affine in control; Attractor x'(t) = Ax(t) + u(t)Bx(t), x(O) = Xo E H. u(t) = -(x(t), Bx(t)), if the following condition is satisfied: (I) (2) 1. Introduction Feedback stabilization of systems that are affine in control is a problem of a great importance in control theory. Several approaches, including Liapunovlike methods, LaSalle's invariance principle and centre Correspondence and offprint requests to: L. Berrahmoune, Departe- ment de Mathematiques, ENS de Rabat BP 5118 Rabat, Morocco. *Emai1: yelboukfaoui@ucam.ac.ma tEmail: erraoui@ucam.ac.ma (etA x, BetA x) = 0 for all t 0 implies x = O. (3) In other words, the set M = {x E H, (etA x, BetA x) = o Vt O} is reduced to zero. The extension of this re- sult to infinite dimensional case, that is, when A is the infinitesimal generator of a Co-semigroup on a real Hilbert space and B is a continuous operator, has been recently the subject of many researches [6,8-10,12]. In particular, in [6] Ball and Slemrod have shown that condition (3) is sufficient for weak stabilizability of system (1) when A is dissipative and B is locally Received 7 June /999. Accepted in revisedform 14 February 2001. Recommended by D. Normand-Cyrot and A. 1sidori ..-.... __ . ,