The dynamic programming method in systems with states in the form of distributions

The problem of optimal control of a system with the initial state in the form of a known distribution function specified on a fixed time segment is considered. To solve this problem, an approach is used in which the state of the system is understood as a coordinate distribution at each instant of time. Analogs of the equations in the Hamiltonian formalism for the problem of minimizing the integral functional are derived. The solution to the problem of optimal control in the closed form for a linear system with an integral quadratic functional is presented.