Local solutions of the Hamilton–Jacobi–Bellman equation for some stochastic problems

Consideration is given to the control problem of motion of a linear oscillator, which is subject to the external Gaussian and Poisson random actions, with the aim to minimize the mean energy with the aid of the external bounded control force. The hybrid solution method is suggested for the solution of the stated problem. This method relies on the search in a portion of the phase space for the exact analytical solution of the appropriate Hamilton–Jacobi–Bellman (HJB) equation and the numerical solution of this equation in the remaining (bounded) portion of the space. It is proved that the found analytical solutions represent the asymptotics of solutions of the Hamilton–Jacobi–Bellman equation. With the aid of the decomposition method, the obtained results are applied to the problem for the suppression with the aid of the actuator of vibrations of an elastic rod (plate) that is found to be under the action of Gaussian random actions. Results of the numerical modeling are given.