The class of optimal control problems is considered in which the terminal point is constrained to lie on some surface in the state space. A computational algorithm is developed which solves the problem for a series of terminal points on the given surface and iterates until the transversality condition is satisfied. An example is considered in which there are several solutions which satisfy the transversality condition, some producing a local minimum of the performance criterion with respect to the variable terminal point and some a local maximum. A sufficient condition for a local minimum is derived.

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