Abstract :
In the last 15 years the problem of stabilizability and stabilization of descriptor systems have received considerable attention. In this paper it is shown that if a descriptor system image exhibits impulsive behavior, then the stability of the closed-loop system is extremely sensitive to small delays. More precisely, if F is the feedback which leads to a stable and impulsive-free closed-loop system, then there exist numbers var epsilonj > 0 and sj ε image with limj → ∞ var epsilonj = 0 and limj → ∝ Re sj = + ∝ and such that the delayed closed-loop system obtained by applying the feedback u(t) = Fx(t − var epsilonj) has a pole at sj. Moreover, if the open-loop system does not have impulsive behavior, the same phenomenon occurs, provided that the spectral radius of the matrix lims → ∝ F(sE − A)−1B is greater than 1. If this spectral radius is smaller than 1, it is shown that the closed-loop stability is robust with respect to small delays.
