Abstract
Consider a system described by the stochastic differential equation $dx = f(t,x,u)dt + \sigma (t,x)dw$ where w is a Wiener process and u lies in a compact set. If solutions are defined in the sense of Girsanov rather than of Itô, then it is natural to work not with the state at time t but rather with the distribution of $x(t)$. It is shown by an extension of the standard argument that if the state reaches a target set, i.e. a specified set of probability distributions, in finite time then there is a first such time. Next it is shown that if the system is linear in u and if $ - 1 \leqq u \leqq 1$, then the attainable distributions arising from bang-bang controls are weakly dense in the set of attainable distributions. Finally some geometric properties of the bang-bang attainable densities are discussed; for example, the exposed points of the attainable densities are the bang-bang attainable densities.
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