Stochastic control modeling for the problem of random motion

The effect of asymptotic suppression of random disturbances is considered using the example of a thermal conductivity equation with variable boundary conditions. Using a precisely solved example, it is shown that a random switching of boundary conditions of the first and second kinds in a one-dimensional finite rod with a non-zero initial condition can lead to a zero solution of the thermal conductivity equation in the limit when the number of switches tends to infinity. This problem is connected with two other problems that are close in formulation: on the distribution of an ensemble of trajectories of a linear dynamic system with random switches and on the construction of a solution to the heat equation with a variable coefficient. The general approach to solving such problems consists in constructing a numerical algorithm that simulates the evolution of the density of the distribution function of the system during random switching at fixed ends, when a new state is known, but the moment of switching is not known. When switching the thermal conductivity coefficient, an instantaneous change in the equation parameter occurs. In a more general approach, the transition is between distributions. The paper provides an example of the application of a numerical algorithm for modeling an ensemble of trajectories to the construction of a distribution for solving equations with random switches. At the same time, two situations are considered – when the distribution of a random parameter (for example, the coefficient of thermal conductivity) is stationary, as well as for the non-stationary case. This approach allows us to numerically determine the range of acceptable variations in the distribution of random parameters, in which the solution of the stochastic equation lies within certain limits, which is of practical interest.