Most of the current work on the maximum principle in control theory, with the exception of Young [11] and McShane [12], deals with necessary conditions that a [any] postulated optimal solution must satisfy. The essential weakness here is that necessary conditions are proved for an often nonexistent solution--Young [11] cites a paradox of Perron to illustrate the logical difficulty this leads to. In partial resolution, Young and McShane first prove existence of relaxed controls and then deduce necessary conditions in the form of the maximum principle for such a solution.
In this paper, we take a strictly computational approach to the problem. We develop a computational procedure for the control problem and the maximum principle essentially "pops out" in the process. The basic idea [1] is easily explained in reference to the simple canonical control problem: T Minimize f g(t; x(t); u(t)) dr, (1.I) 0 Yc(t) = f(t; x(t); u(t)); x(O) -~ xl , x(T) = x 2 given (1.2) and the controls u(t) [Lebesgue measurable] are constrained to belong to U compact for each t [a.e.]. We eliminate the time-consuming phase of solving the dynamic equation by introducing the nondynamic problem for each E > 0: Minimize f~ [ ~--~-H x(t)--f(t;x(t),u(t))]]'~-g(t;x(t);u(t)] dt (1.3) over the class of state functions x(t), absolutely continuous with x(O) ~ Xl, x(T) = x 2 and over the class of controls as before. [Note that x(t) is no longer constrained to