  In this paper, we analyze the behavior of stochastic approximation schemes
with set-valued maps in the absence of a stability guarantee. We prove that
after a large number of iterations if the stochastic approximation process
enters the domain of attraction of an attracting set it gets locked into the
attracting set with high probability. We demonstrate that the above result is
an effective instrument for analyzing stochastic approximation schemes in the
absence of a stability guarantee, by using it obtain an alternate criteria for
convergence in the presence of a locally attracting set for the mean field and
by using it to show that a feedback mechanism, which involves resetting the
iterates at regular time intervals, stabilizes the scheme when the mean field
possesses a globally attracting set, thereby guaranteeing convergence. The
results in this paper build on the works of V.S. Borkar, C. Andrieu and H. F.
Chen , by allowing for the presence of set-valued drift functions.
