Physical or geometrical meanings of optimum trajectory, optimum control and impulse function
p are clarified to some extent by introducing the notion of maximum travel surface into the exposition of maximum principle. In the problem where the time to be controlled is specified and the terminal point is unknown, terminal point in
n+1-space is determined as the tangent point of maximum travel surface with time
T constructed with respect to initial point to the hyperplane with the direction specified by Pontryagin function. The locus fo contact point of maximum travel surface with time
t constructed with respect to initial point with one with time
T-
t constructed with respect to terminal point is really optimum trajectory. The force to be applied to the controlled system in order not to let the phase point escape out of this trajectory is optimum control. From these facts, it is concluded that impulse function
p is no more than the gradient vector of maximum travel surface at the contact point. Although the convexity of maximum travel surface in the special problem is affirmed, many questions concerning to convexity of maximum travel surface as well as its smoothness still remain unsolved. It is conjectured that when one of maximum travel surface has corner at the contact point, the other maximum travel surface must be smooth at that point. The solution of minimum time control problem is extremely clarified by analyzing the problem in
n-space through maximum travel surface concept.
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