Chabanyuk Y. Stochastic approximation in evolution systems withe Markov and semi-Markov switchings.

The dissertation is devoted to the research of stochastic system stability with semi-Markov switchings, convergence of continuous and jumping stochastic approximation procedures in the case, when the regression function depends on outgoing conditions, the asymptotic normality of the stochastic approximation procedure in the Markov and semi-Markov processes. There is obtained the technique of the research of stochastic system stability in the averaging scheme and in the diffusion approximation scheme in semi-Markov environment through Lyapunov's function for averaging system in the dissertation. The theorems of continuous procedure of the stochastic approximation for the regression function convergence, which depends directly on outside environment, meaning has the following form C (u, x), where the variable x is defined by such an environment, are proved. In the case, when outside environment is determined by uniformly ergodic Markov process, the sufficient conditions of procedure convergence in the averaging scheme and in the diffusion approximation scheme are obtained in the terms of existence of Lyapunov's function for averaged system with stationary distribution of Markov process. The small parameter of eps. series usage gave an opportunity to modify the solution of singular perturbation problem for obtained asymptotic splitting generators.