Stability of Linear Continuous-Time Systems With Stochastically Switching Delays

Necessary and sufficient conditions for the stability of linear continuous-time systems with stochastically switching delays are presented in this paper. It is assumed that the delay random paths are piece-wise constant functions of time where a finite number of values may be taken by the delay. The stability is assessed in terms of the second moment of the state vector of the system. The solution operators of individual linear systems with constant delays, chosen from the set of all possible delay values, are extended to form new augmented operators. Then for proper formulation of the second moment in continuous time, tensor products of the augmented solution operators are used. Finally the finite-dimensional versions of the stability conditions, that can be obtained using various time discretization techniques, are presented. Some examples are provided that demonstrate how the stability conditions can be used to assess the stability of linear systems with stochastic delay.