Robust stability, stabilization and ℋ∞ control of time-delay systems with Markovian jump parameters

In this paper, the problems of stochastic stability and stabilization for a class of uncertain time-delay systems with Markovian jump parameters are investigated. The jumping parameters are modelled as a continuous-time, discrete-state Markov process. The parametric uncertainties are assumed to be real, time-varying and norm-bounded that appear in the state, input and delayed-state matrices. The time-delay factor is constant and unknown with a known bound. Complete results for both delay-independent and delay-dependent stochastic stability criteria for the nominal and uncertain time-delay jumping systems are developed. The control objective is to design a state feedback controller such that stochastic stability and a prescribed ℋ∞-performance are guaranteed. We establish that the control problem for the time-delay Markovian jump systems with and without uncertain parameters can be essentially solved in terms of the solutions of a finite set of coupled algebraic Riccati inequalities or linear matrix inequalities. Extension of the developed results to the case of uncertain jumping rates is also provided.