Using first integrals to estimate limiting possibilities of optimal control systems

In a continuation of our previous investigations devoted to the question of using first integrals in problems of the optimal control of dynamical systems, the optimal control problem for the motion of a multidimensional non-linear system over a given time interval with fixed endpoints for the phase trajectory is considered. The quality of the control is estimated by a functional in the form of a definite integral of a weighted sum of squares of the components of the controlling forces. In a number of cases such a functional gives an estimate of energy losses during control, and the corresponding variational problem is called optimal for minimum energy loss. Based on the use of first integrals of the free equations of motion a method is developed to find the upper limit of the least necessary energy loss to displace the controlled non-linear dynamical system from a given initial phase state to a given final state in a given time. The efficiency of the method is illustrated by examples, including the solution of the problem of estimating the limiting possibilities of an energetically optimal control system for controlling the motion of an artificial satellite in a gravitationally attractive Newtonian central field