Dynamical Problems of Optimal Control

Abstract

The main contents of the present chapter are recurrence algorithms for obtaining numerical solutions of dynamic problems of optimal control. Two new paths will be explored. These are based on the fundamental Lagrange - Hamilton principles of analytical dynamics and on the idea of modelling of differential equations of controlled motion by generalized monogeneous force fields. The first path is based on the idea of shifting nonholonomic elastic constraints the essence of which was presented in Sec.17.5. In constrained optimization problems (see Chapter 6), the optimal shifts of constraints, whose models are strong fields, are of constant magnitude and are determined by the intensity in the equilibrium state of the suitable force fields representing the constraints. In contrast, in dynamical problems of optimal control the constraints are nonholonomic and their shifts are functions of time and are also connected in a natural manner with Lagrange multipliers. The idea of modelling of nonholonomic constraints (differential equations of controlled motion) enables us to construct a functional, analogous to the Hamilton action, and transform the problem of optimal control to the classical problem of the calculus of variations.