Abstract
This paper deals with the solution of optimum control problems, using Pontryagin's maximum principle. Since an optimum control system which completes the transient response in minimum time behaves like a relay servo, the important task of the controller is to reverse the sign of the manipulated variable at the proper instants according to the operating condition of the system. A set of points at which the sign of the manipulated variable is to be reversed forms a switching surface in the phase space. In this paper the optimum switching surface of a linear third-order system is investigated. Also dealt with is the optimum switching curve of a non-linear control system governed by van der Pol's equation with an additional forcing term.
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