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On the solutions of a stochastic control system
Duncan, Tyrone E. Varaiya, Pravin
Duncan, Tyrone E.
Varaiya, Pravin
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Abstract
The control system considered in this paper is modeled by the stochastic differential
equation
dx(t, to) f(t, x(., o), u(t, to)) dt + dB(t, to),
where B is n-dimensional Brownian motion, and the control u is a nonanticipative functional of
x(., to) taking its values in a fixed set U. Under various conditions on f it is shown that for every
admissible control a solution is defined whose law is absolutely continuous with respect to the Wiener
measure #, and the corresponding set of densities on the space C forms a strongly closed, convex subset
of LI(C, I). Applications of this result to optimal control and two-person, zero-sum differential
games are noted. Finally, an example is given which shows that in the case where only some of the
components of x are observed, the set of attainable densities is not weakly closed in LI(C, t).
Description
This is the published version, also available here: http://www.dx.doi.org/10.1137/0309026.
Date
1971-01-01
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Society for Industrial and Applied Mathematics
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Citation
Duncan, Tyrone E. "On the solutions of a stochastic control system." SIAM J. Control. (1971) 9, 3. 354-371. http://www.dx.doi.org/10.1137/0309026.