Finite Dimensional Approximations of Doubly Bezout Factors

In this note, a sequence of finite dimensional approximations is constructed for semigroup systems with bounded input and output operators. First, it is shown that the doubly Bezout factorization of the approximating systems converge to those of the semigroup systems correspondingly. Then it is revealed that the unstable parts of the semigroup system will be recovered by the finite dimensional approximations when their dimensions are sufficiently high. The problem of stabilization by feedbacks output injections designed according to finite dimensional approximations investigated. Finally, it is proved that optimally robust controllers of approximating systems stabilize semigroup system.