Fréchet differentiability and optimal control

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Abstract
It is shown that in the standard control problem the trajectory is, in general, a continuously Fréchet differentiable function of the control, the space of admissible controls being given the uniform norm. A formula is given for the Fréchet derivative. It follows from this that a general cost functional on the system is a Fréchet differentiable function of the control. As a preparation for work on numerical methods it is shown that the same things are true for neutral time-lag systems, and a formula is given for the derivative of the cost functional. For simplicity and greater possible usefulness, the main results are stated and proved for piecewise continuous controls ; it is shown how they may be extended to measurable controls, with essentially the same proofs. Another extension, to Banach space-valued controls and trajectories, is given.

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