  An optimal control for a dynamical system optimizes a certain objective
function. Here we consider the construction of an optimal control for a
stochastic dynamical system with a random structure, Poisson perturbations and
random jumps, which makes the system stable in probability. Sufficient
conditions of the stability in probability are obtained, using the second
Lyapunov method, in which the construction of the corresponding functions plays
an important role. Here we provide a solution to the problem of optimal
stabilization in a general case. For a linear system with a quadratic quality
function, we give a method of synthesis of optimal control based on the
solution of Riccati equations. Finally, in an autonomous case, a system of
differential equations was constructed to obtain unknown matrices that are used
for the building of an optimal control. The method of a small parameter is
justified for the algorithmic search of an optimal control. This approach
brings a novel solution to the problem of optimal stabilization for a
stochastic dynamical system with a random structure, Markov switches and
Poisson perturbations.
