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This paper studies an optimal control problem for continuous-time stochasticsystems subject to reachability objectives specified in a subclass of metricinterval temporal logic specifications, a temporal logic with real-timeconstraints. We propose a probabilistic method for synthesizing an optimalcontrol policy that maximizes the probability of satisfying a specificationbased on a discrete approximation of the underlying stochastic system. First,we show that the original problem can be formulated as a stochastic optimalcontrol problem in a state space augmented with finite memory and states ofsome clock variables. Second, we present a numerical method for computing anoptimal policy with which the given specification is satisfied with the maximalprobability in point-based semantics in the discrete approximation of theunderlying system. We show that the policy obtained in the discreteapproximation converges to the optimal one for satisfying the specification inthe continuous or dense-time semantics as the discretization becomes finer inboth state and time. Finally, we illustrate our approach with a robotic motionplanning example.

Computational methods for stochastic control with metric interval temporal logic specifications