Optimal programmed control in energy minimisation problem

Abstract

This paper is devoted to the construction of optimal programmed control of the motion of a material point in a medium with a random coefficient of resistance in a homogeneous field of gravity of the Earth. The control objective is to minimise the mathematical expectation of the total mechanical energy of the material point. The problem is solved by two methods proposed by the author in previous works. Both methods assume approximation of the initial stochastic problem by deterministic optimal control problems. In the first case, the transition to approximate deterministic problems is carried out by replacing the parameters representing continuous random variables by discrete-type random variables converging in distribution to the original continuous ones (the method of distribution discretisation). The meaning of such a replacement is that the resulting approximate problem can be considered no longer as a stochastic, but as a deterministic optimal control problem, and can be solved, accordingly, with the help of known standard methods. In the second case, the initial stochastic dynamic system as a result of averaging its equations is replaced first by an infinite system and then, after zeroing all moments of sufficiently high order, by a finite system of equations for mixed moments of the solution and a random parameter (method of moments). The control functions and the optimal values of the quality functional obtained by the two methods were found to be almost identical, and both methods showed good convergence.