A class of problems that has received considerable attention in recent years from both control theorists and engineers is the following:

Given **ẋ** = **Fx** + **d**u, **x**(0) = **c**Determine |u(t)| ≤ 1 such that **x**(T) = 0 and **x**(t) ≠ 0for 0 ≤ t < T and where T is a minimum (P-1)

A related and perhaps more practical class of problems can be stated as

Given **ẋ** = **Fx** + **d**u, **x**(0) = **c**Determine |u(t)| ≤ 1 such that ‖**x**(T)‖^{2}**P** is a minimum for given T (P-2)

Although a considerable amount of effort has been expended on (P-1), and to a lesser extent on (P-2), yet computational techniques which enable one to solve numerically the above problems are still lacking except in restricted cases [7, 8]. This paper presents such a technique which completely solves this problem by successive approximation. The convergence of this solution is proved, and it is shown to satisfy all known properties of the problems.