Abstract :
New sufficient conditions for functions space controllability and hence feedback stabilizability of linear retarded systems are presented. These conditions were obtained by treating the retarded system as a special case of an abstract equation in Hilbert space Rn ?? L2([-h,0],Rn), (denoted as M2). For systems of type x(t)= A0x(t) + A1x(t-h) + Bu(t), it is shown that most of controllability properties are described by a certain polynomial matrix P(??), whose columns can be generated by an algorithm computing A0 iB, A1 iB and mixed powers of A0 and A1 multiplied by B. It is shown that the M2- approximate controllability of the system is guaranteed by certain triangularity properties of P(??). By using the Luenberger canonical form, it is shown that the system is M2-approximately controllable if the palr (A1,B) is controllable and if each of the spaces spanned by columns of [B,A1B, ..., A1 jB], j=0...n-1, is invariant under transformation A0. Other conditions of this type are also given. Since the M2-approximate controllability implies controllability of all the eigenmodes of the system, the feedback stabilizability with an arbitrary exponential decay rate is guaranteed under hypotheses leading to M2-approximate controllability. Some examples are given.
