Best Endpoint Control of Noisy Systems

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Abstract
The endpoint control problem for noisy systems, as formulated in terms of dynamic programming (Bellman 1957), loads to a control policy which requires complete feedback information at every time when control action is to be taken. The problem formulation applies conceptually only to systems making discrete jumps at discrete times, although the functional equations solution can in certain cases (in particular where the random disturbances to the system equations appear as an additive Gaussian white noise) be taken to the limit of continuous time (Florentin 1961) The present work investigates an alternate formulation of the problem that aims in the first instance at a feed forward control schedule, although it can in principle be used to obtain feedback control policies as well. The policies obtained in this way differ in general from those arising from the formalism of dynamic programming. This formulation applies equally well to systems evolving continuously in time, and, under certain restrictive conditions (where the random disturbances to the system equations take the form of an additive Gaussian white noise), the problem can be solved by what are essentially classical methods of the calculus of variations. This mathematical solution is presented here, and carried out explicity in a simple case for comparison with the dynamic programming solution There are engineering situations where the feed forward solution itself has an intrinsic interest. A case in point, drawn from the chemical industry, is the design of temperature control schedules for batch reaction systems subject to random shocks, where the reaction environment makes interim measurements of the state variables (composition) a matter of great difficulty. But the procedures presented here call for such massive computation in practical cases that it is difficult to see how one could justify their routine application.

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