Управление системой, обеспечивающее периодическое в среднем решение

Abstract

The results of this paper have been presented at the 11-th Russian Multiconference on control problems (St. Petersburg, 2–4 October 2018). The formula for the expectation of solutions of a linear diffusion equation with one spatial variable and random coefficients has been obtained. For this we find firstly an auxiliary non-random differential equation with ordinary and variational derivatives and a determinate initial condition. For the particular case of the equation under consideration, namely the heat equation, in which the coefficient in front of an unknown function is Gaussian or uniformly distributed, conditions for the existence of mean periodic solutions are given. If we consider the non-homogeneity as a control function, then obtained results can be interpreted as finding a control that ensures the periodicity of the expectation of a solution.