Semistability of retarded functional differential equations

In this paper, we develop a semistability analysis framework for retarded functional differential equations (RFDE) having a continuum of equilibria with time-varying parameters and delays with applications to stability analysis of multiagent dynamic networks with consensus protocols in the presence of unknown heterogeneous time-varying delays and parameters along the communication links. We show that for such a retarded functional differential equation, if the system asymptotically converges to an autonomous functional differential inclusion with constant time-delays and this new system is semistable, then the original retarded functional differential equation system is semistable, provided that the delays are just bounded, not necessarily differentiable. In proving our results, we extend the limiting equation approach to the retarded functional differential equation systems and also develop some new convergence results for functional differential equations and differential inclusions.