Systems with gamma-distributed delays: A Lyapunov-based analysis

In the present paper, sufficient conditions for the exponential stability of linear systems with gamma-distributed delays are presented. Such systems arise in populations dynamics, in traffic flow models, in networked control systems and in other engineering problems. Our main challenge is the stability conditions, where the delay is stabilizing, i.e. the corresponding system with the zero-delay is not asymptotically stable. The results are derived by using augmented Lyapunov functionals, were we generalize the earlier results of [1] and [9], regarding distributed delays and finite constant kernels, to the infinite delay case by extending the corresponding Jensen's integral inequalities and Lyapunov-Krasovskii constructions. Polytopic uncertainties in the system matrices can be easily included in the analysis. Numerical examples illustrate the efficiency of the method. Thus, for the traffic flow model on the ring, where the delay is stabilizing, the resulting stability region almost coincides with the theoretical one found in [11] via the frequency domain analysis.