Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems

New sufficient conditions for function space controllability and hence feedback stabilizability of linear retarded systems are presented. These conditions were obtained by treating the retarded systems as a special case of an abstract equation in Hilbert space
R^{n}\times L_{2}([- h, 0], R^{n})
(denoted as
M_{2
}). For systems of type
\cdot{x}(t)=A_{0}x(t)+A_{1}x(t-h)+Bu(t)
, it is shown that most of controllability properties are described by a certain polynomial matrix
P(\lambda)
, whose columns can be generated by an algorithm comparing
A_{0}^{i}B,A_{0}^{i} B
and mixed powers of A
0
and A
1
multiplied by
B.
It is shown that the M
2
-approximate controllability of the system is guaranteed by certain triangularity properties of
P(\lambda)
. By using the Luenberger canonical form, it is shown that the system is M
2
-approximately controllable if the pair
(A_{1},B)
is controllable and if each of the spaces spanned by columns of
[B,A_{1}B,... ,A_{1}^{j}B], j=O...n-1
, is invariant under transformation A
0
. Other conditions of this type are also given. Since the M
2
-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 M
2
-approximate controllability. Some examples are given.