The problem of synthesis of the optimal control for a stochastic dynamic system of a random structure with Poisson perturbations and Markov switching is solved. To determine the corresponding functions for Bellman functional and optimal control the system of ordinary differential equation is investigated.

An optimal control problem for systems of stochastic differential-functional linear equations with past history and Poisson switchings is formulated. The Bellman equation is solved for this problem.

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Abstract. Sufficient conditions for the mean square stability of solutions of linear stochastic differential-functional Itô-Skorokhod equations with unbounded aftereffect are obtained in the paper. The critical case is also studied. Asymptotic mean square stabilityLet (Ω, F, P) be a probability space and Let {x(t) ≡ x (t, ω)} ⊂ R be a stochastic process defined for t ≥ 0 by the stochastic differential-functional Itô-Skorokhod equationPoisson measure in R with parameter tΠ (A) ≡ E {ν (t, A)} where {w(t)} and {v (t, A)} are independent and F t -measurable for t ≥ 0.The coefficients a, b, and g are linear functionals for any t ≥ 0 defined on R + × D 0 , R + × D 0 , and R + × D 0 × R, respectively. We treat D 0 as a metric space with the Skorokhod metric ρ D (see [4, Chapter VI,§5]).To facilitate the discussion of the behavior of stochastic processes {x (t)} ⊂ R without discontinuities of the second kind, a simpler metric is often considered (see [3]). This metric is generated by the seminorm

The Markov properties of the solutions of Ito-Skorokhod stochastic functional-differential equations (SFDEs) with entire prehistory are considered, the concept of a weak infinitesimal operator is introduced for a Markov process that is a solution of an SFDE, and the strong solution of the SFDE is analyzed for stability.The concept of Lyapunov stability, namely, the method of Lyapunov functions (Lyapunov-Krasovskii functionals)[10] is the most constructive and universal approach to the stability analysis of deterministic systems. The main advantage of this method is that it allows concluding whether a system whose analytical solution cannot be found by the direct integration of the differential equations is stable.In developing the theory of stability of stochastic systems, the following aspects are fundamentally important: · considering systems that possess the Markov property; · the concept of strong probabilistic stability, which makes it possible to analyze the asymptotic behavior of realizations of a random solution-process of the corresponding stochastic equation; · using the method of functions (functionals), namely, the concept of a derivative for deterministic systems or a weak infinitesimal operator for stochastic dynamic systems which can be computed without the need to integrate the equation of perturbed motion.