Condition (0.2) is strictly linked to linear quadratic optimal control and has to be assumed in order to deal with “infinite horizon” problems (see [8], [3]). In the present paper the control theory techniques are widely used and large part of the work deals with the manipulation of the Riccati and algebraic Riccati equation relative to the dynamic system described by (0.1) (see (T l£ . ) and (A7£.£))*

Some work on the stabilizability of equation (0.1) can be found in [4] where a condition on (^ 4, 2?, C ) implying (0.2) is given. In [9], although most of the attention is devoted to the stabilizability of a stochastic system with arbitrary noise intensity, a criterion for the stabilizability of (A , B , C) (involving the computation of the solution of a Riccati Equation) is given in the special case in which the image of B and the image of C have dimension

one. Finally some results (sufficient conditions for the /-stabilizability of (A, B } C ) ) in the infinite dimensional case can be found in [7].