Abstraction-based Synthesis of Continuous-Time Stochastic Control Systems

This paper proposes a scheme for the construction of discrete-time finite abstractions (a.k.a. finite Markov decision processes) from continuous-time stochastic control systems and quantifying their probabilistic distances. The scheme is based on the notion of stochastic simulation functions enabling us to relate continuous-time stochastic systems with their discrete-time counterparts. Accordingly, one can employ discrete-time abstract systems as substitutions of the continuous-time ones in the controller design process with guaranteed error bounds on their output trajectories. In the first part of the paper, we quantify the distance in probability between original continuous-time stochastic control systems and their discrete-time (finite or infinite) abstractions. In the second part of the paper, we construct finite abstractions together with their corresponding stochastic simulation functions for a particular class of stochastic affine systems having some stability property. We demonstrate the effectiveness of the proposed results by applying our approaches to the temperature regulation in a building of two adjacent rooms and constructing a discrete-time abstraction from its original continuous-time dynamic. We employ the constructed discrete-time abstraction as a substitute to synthesize policy regulating the temperature of rooms for a bounded time horizon.