Asymptotic Analyses for Complex Evolutionary Systems with Markov and Semi-Markov Switching Using Approximation Schemes

The thesis contains some applications of the developed methods to the analysis of the model of counteraction to information attacks. The monograph includes thefollowing new results:
– under the Levy approximation conditions, a limiting generator was constructed and an asymptotic behavior analysis of stochastic evolution systems with impulse perturbation and Markov switches was carried out;xii Asymptotic Analyses for Complex Evolutionary Systems – under the Poisson approximation conditions, an asymptotic analysis of the behavior of stochastic evolution systems with impulse disturbance and Markov switches was carried out;
– for a Poisson approximation, the boundary process contains two components: deterministic shift and Poisson jump component. In contrast, for a Levy approximation, the Levy limiting process contains three components: deterministic
shift, diffusion time and Poisson jump component. In this case, the function that defines a deterministic shift is determined by frequent small jumps of the process; the function, which sets the diffusion component, is determined by frequent large jumps of the process; finally, the high jumps of the boundary process are set by the measure of rare large jumps of the border process;
– an asymptotic dissipative analysis of stochastic differential equations with impulse effects under the conditions of nonclassical approximation schemes (Levy and Poisson) was carried out; – the conditions of weak convergence of the diffusion transfer process with Markov switches and the control of the function of the quality criterion with the
equilibrium point for which the stochastic approximation procedure was constructed in the series scheme;
– for the process of transfer with Markov switches and control under the conditions of existence of a single point of equilibrium of the quality criterion, a normalized process was constructed and its asymptotic normality was established in the Ornstein– Uhlenbeck process when the transfer process changed under the influence of the Markov switching on the trajectory of a new evolution from the state in which it was at the moment of switching;
– for the stochastic approximation procedure, a generalization to the case of the extrinsic environment dependent regression function with Markov and Semi-Markov switches is considered. Sufficient stochastic convergence conditions for Markov switches were obtained for the stochastic optimization procedure.