Time-Optimal Control of Second Order Systems with Parameter Variations

Abstract

Second order control systems having two system prameters as control inputs, x+2u₃x+u₂x=0, are analysed for the optimal control problems to minimize the time from an arbitrary initial point to the region on a circle around the origin of the state space.
Applying Pontryagin's maximum principle to this problem, a unique solution is not necessarily obtained.
This paper presents the conditions of the unique solution and a method of obtaining the optimal trajectory when the unique solution is unobtainable and in addition, using analogue computer, the importance of a boundary line named “Separatrix” is shown in getting the optimal trajectory, and it is pointed out that the switching lines obtained by Pontryagin's maximum principle correspond to the extremum of the functional.