This paper presents an extension to the problem discussed in previous papers by Irmgard Flu¨gge-Lotz and Mih Yin.2, 3 In the full third-order problem it was found desirable to modify the iteration procedure used in the periodic approximation method that was developed in solving the optimum problem of the second-order, velocity-controlled system.2 A number of examples illustrate the new method. The examples were first constructed in reverse time by employing the true optimum switching times satisfying the optimum switching function, equation (6), which was developed by the methods of Pontrjagin.2 The final values of error and error derivatives which are obtained give the initial values for testing the approximate method. Periodic approximations of the optimum switching times are then obtained. It was found that for less than three switchings the approximate method leads to identical results. Reasonably small differences occur when three or more switchings are needed to reduce the initial disturbance to zero.

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