This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the state. x(t), the control u(t), and the parameter pi so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, While the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t), pi obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The stepsize of the gradient phase is de: ermined by a one-dimensional search on the augmented functional J, and the stepsize of the restoration phase by a one-dimensional search on the constraint error P. If alpha(g) is the gradient stepsize and alpha(r) is the restoration stepsize, the gradient corrections are of O(alpha(x)) and the restoration corrections are of O(alpha(r)alpha(2)(g)). Therefore. for alpha(g) sufficiently small the restoration phase preserves the descent property of the gradient phase: the functional (I) over cap at the end of any complete gradient-restoration cycle is smaller than the functional I at the beginning of the cycle. To facilitate the numerical solution on digital computers, the actual time theta is replaced by the normalized time t, defined in such a way that the extremal arc has a normalized time length Delta t = 1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time tau at which the terminal boundary is reached is regarded to be a component of the parameter pi being optimized. The present general formulation differs from that of Ref. 4 because of the inclusion of the nondifferential constraints to. be satisfied everywhere over the interval 0 <= t <= 1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) problems involving state equality constraints can be reduced to the present scheme through suitable transformations, and (iii) problems involving inequality constraints can be reduced to the present scheme through suitable transformations. The latter statement applies, for instance, to the following situations: (a) problems with bounded control. (b) problems with bounded state, (c) problems with bounded time rate of change of the state, and (d) problems where some bound is imposed on an arbitrarily prescribed function of the parameter, the control, the state, and the time rate of change of the state. Numerical examples are presented for both the fixed-final-time case and the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.
