About numerical approximations of a stabilizing distributed delay control law

Since the end of the seventies, several authors proposed to use a distributed delay in the control law of time-delay systems. Some recent publications showed that the implementation of such laws is not always safe with respect to the stability of the closed-loop system, as we recall in the first part of this paper limited to the scalar case of systems with delayed input. In the second part, we give a condition of stabilizability issued from complex analysis considerations. The last part of this work is devoted to the expression of conditions on the parameters of the system that match the condition given in the previous part. We exhibit one necessary and one sufficient condition of stabilizability for system with delayed input. 1 Motivation In 1979, A. Manitius and A. Olbrot [2] proposed a method of pole placement for time-delay systems by using a distributed delay in the control law, i.e. a finite integral over the past values of the state or the command of the system. For example, a system with a pure delay on its input : _ x(t) = ax(t) + u(t h) (1) would be controlled by a law of the form