The heredity shift operator in descent approach to the optimal periodic hereditary control problem

Abstract

A new computational approach to the optimal periodic hereditary control problem is proposed. The description of the periodic hereditary process in the equivalent integral form using the forward heredity shift operator is considered. The particular form of such a model reduced to a fixed point of a mapping defined on the control horizon is advantageously exploited in computational algorithms. The normal Lagrange functional approach is applied to derive the integral adjoint equation with the backward heredity shift operator, and to obtain the gradient formula of the goal functional with respect to the control. The descent optimization method, employing the Newton algorithm for solving the periodic integral state equation in the roof basis, and the projected gradient method in the control space with the step basis, is depicted. To verify the properness of the problem, and to estimate the best operation period the /7 test specialized to the case of many lumped and distributed delays is obtained. The computer implementation of the method is performed in the Mathematica system, and verified on examples of periodic control problems for ecological processes.