Optimal trajectories and linear control of nonlinear systems

Abstract

The previously reported successive-approximations scheme, based on the adjoint system, for the integration of the nonlinear variational two-point boundary-value problem is extended here to problems in which the initial and terminal boundary conditions are general functions of the problem variables, thus permitting the treatment of a general Mayer problem. Inequality constraints, which may, but need not, explicitly depend on the control, are included. The linear relationship (a Green's theorem) used in the successive approximations procedure also yields the linear feedback control law in the neighborhood of an optimal nominal trajectory (neighboring extremal control). The adjoint system is equivalent to previous analyses based on the second variation of the calculus of variations, and the feedback control scheme derived here is similar. The present method of constructing the feedback gains for certain neighboring extremal control problems involves less computation than previous methods. Numerical examples of both the optimization and control schemes are presented for planar rocket maneuvers outside the earth's atmosphere.