A direct second-variational method for free-endpoint, free-time optimal control problems by Newton's method in function space†

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Abstract
A direct second-variational method is proposed for solving free-time, free-endpoint optimal control problems. By introducing a time-scaling factor and a reduced-time coordinate, the original free-endpoint problem is transformed into a fixed-time problem in the reduced-time domain. In this framework, Newton's method is applied to the optimality criterion and the transversality condition, in order to arrive at a scheme for simultaneous correction in the control policy and in final time. The resulting algorithm does not involve a Riccati transformation nor requires the solution of other sensitivity equations, and is therefore a decided improvement over existing methods. For illustrative purposes the method is applied to the temperature optimization of a tubular reactor of unspecified length.