Section: New Results

Some remarks on the Walton and Marshall method for neutral delay systems

Participants : Catherine Bonnet, Islam Boussaada, Le Ha Vy Nguyen.

The Walton and Marshall method allows to determine stability windows of delay systems of the retarded and neutral type. We noticed that some delay systems of the neutral type do not behave as claimed in [59] and analyzed carefully the position of the poles of such systems in the right half-plane. We have considered delay systems with characteristic equation being a quasi-polynomial with one delay and polynomials of degree one. It is shown that for a subclass of systems which have a chain of poles clustering the imaginary axis by the left, the procedure of Walton and Marshall fails: we prove the existence, for an infinitesimally small delay, of a positive real pole at infinity. This real pole is then proved to be the unique pole of the system in the closed right half-plane for all values of the delay. Some numerical examples illustrate the results [39]. We are currenlty extending those results to the general case of polynomials of degree $n>1$.