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In this paper the authors present two examples of the application of the standard method of solving the optimal control problem, which they have been developing, for the time-optimal control problem of the second order systems; x+2u3x+u2x=u1 and x+2u3x+x=u1. They consider first the relation between the dimensions of control vector and state vector, and show that it is unreasonable to consider three control variables in the second order controlled system in the course of applying the maximum principle, and how these unreasonable problems can be dealt with. It is also shown that a system in which parameters are used as control variables is not a very special kind of system but it is only a type of nonlinear control systems. In case of the system x+2u3x+x=u1, reverse time trajectories, which are constructed through the application of the standard method, intersect with each other in the limited region of x1x2-plane, and this fact has been giving one an impression that this problem is a very special one. The authors concluded that this phenomenon has come from the fact that maximum principle is a necessary condition for optimality and it does not give a sufficient condition. Complete solution of this problem is also given. Feedback control scheme, which is the main goal of the standard method, is obtained for both of the above second order systems.

Minimum Time Control of Second Order Systems (treated by Masubuchi and Others) through Standard Method