Infinitesimal methods in control theory: Deterministic and stochastic

Abstract

In Part I, methods of nonstandard analysis are applied to deterministic control theory, extending earlier work of the author. Results established include compactness of relaxed controls, continuity of solution and cost as functions of the controls, and existence of optimal controls. In Part II, the methods are extended to obtain similar results for partially observed stochastic control. Systems considered take the form:dx, = f(t, x, y, u(t, y)) dr+ g(t, x, y, u(t, y)) db,, dy, = f (t, x, y, u(t, y)) dt+ g(t, y) d/~, where the feedback control u depends on information from a digital read-out of the observation process y. The noise in the state equation is controlled along with the drift. Similar methods are applied to a Markov system in the final section. (1980). 49A10, 49A60, 93E20, 03H05. AMS (MOS) subject classifications

Optimal control for Navier-Stokes equations

“…(In [7] it is shown how this can be defined explicitly using Loeb measures.) Note that for a control (f el) we have °(*(/:') = (f.…”

“…The task then is to see whether Ojsf can be transformed into a standard optimal control of some kind. For finite dimensional DEs and SDEs this idea was developed in [7] and related papers.…”

“…As is customary in optimal control theory, it is often necessary to extend this class to its natural completion, which is the class of generalized or relaxed controls. These, and the topology on them are as defined thus (as in [7]). The compact metric space M is fixed for the whole paper.…”

“…The fundamental results about controls that we shall use are summarized below; for details consult for example [7], [15], [2] or [20] (noting that a relaxed control is a particular case of a Young measure).…”

“…The methods involve the Loeb space techniques that were employed in [4,6] to solve the stochastic Navier-Stokes equations with general force and multiplicative noise (that is, with noise g{s^u{s)) involving feedback of the solution n), combined with the nonstandard ideas used earlier in the study of optimal control of finite dimensional equations (see [7] for example). For the 3-d case it is necessary to utilize the idea of approximate solutions developed in [11] for the study of attractors.…”