A remark on the Bellman principle of optimality

The group-theoretical approach based on joint use of the Lie-Ovsyannikov infinitesimal apparatus (1–3) and the Noether theory of invariant variation problems (4) is suggested for the problem of synthesis of optimum control. Up to now the ideas of the Noether theory were used on the theory of optimum control for the problems of programmed control only and considered in a small number of publications; as for the problem of synthesis, this paper is the first in this area. In this paper a corollary of the Bellman equation is obtained in the form of a linear partial differential equation. The use of the equation simplifies the problem of construction of synthesizing controls (note, that the equation is correct both in the case when no restrictions are imposed on the vector of control, and in the case when the values of the vector belong to a bounded closed domain. An example illustrating the technique of use of the Lie-Ovsyannikov infinitesimal apparatus for reduction of the corresponding Bellman equation is presented.