  This paper studies an optimal control problem for continuous-time stochastic
systems subject to reachability objectives specified in a subclass of metric
interval temporal logic specifications, a temporal logic with real-time
constraints. We propose a probabilistic method for synthesizing an optimal
control policy that maximizes the probability of satisfying a specification
based on a discrete approximation of the underlying stochastic system. First,
we show that the original problem can be formulated as a stochastic optimal
control problem in a state space augmented with finite memory and states of
some clock variables. Second, we present a numerical method for computing an
optimal policy with which the given specification is satisfied with the maximal
probability in point-based semantics in the discrete approximation of the
underlying system. We show that the policy obtained in the discrete
approximation converges to the optimal one for satisfying the specification in
the continuous or dense-time semantics as the discretization becomes finer in
both state and time. Finally, we illustrate our approach with a robotic motion
planning example.
