STABILITY OF CONTROLLED STOCHASTIC DYNAMIC SYSTEMS OF RANDOM STRUCTURE WITH MARKOV SWITCHES AND POISSON PERTURBATIONS

Lyapunov’s second method is used to study the problem of stability of controlled stochastic dynamical systems of random structure with Markov and Poisson perturbations. Markov switches reflect random effects on the system at fixed points in time. Poisson perturbations describe random effects on the system at random times. In both cases there may be breaks in the phase trajectory of the first kind. The conditions for the coefficients of the system are written, which guarantee the existence and uniqueness of the solution of the stochastic system of a random structure, which is under the action of Markov switches and Poisson perturbations. The differences between these systems and systems that do not contain internal perturbations in the equation, which cause a change in the structure of the system, and external perturbations, which cause breaks in the phase trajectory at fixed points in time, are discussed. The upper bound of the solution for the norm is obtained. The definition of the discrete Lyapunov operator based on the system and the Lyapunov function for the above-mentioned systems is given. Sufficient conditions of asymptotic stochastic stability in general, stability in l.i.m. and asymptotic stability in the l.i.m. for controlled stochastic dynamic systems of random structure with Markov switches and Poisson perturbations are obtained. A model example that reflects the features of the stability of the solution of a system with perturbations is considered: the conditions of asymptotic stability in the root mean square as a whole are established; the conditions of exponential stability and exponential instability are discussed. For linear systems, the necessary and sufficient stability conditions are determined in the example, based on the generalized Lyapunov exponent.