Formal and Efficient Synthesis for Continuous-Time Linear Stochastic Hybrid Processes

Stochastic processes are expressive mathematical tools for modeling real-world systems that are subject to uncertainty. It is hence crucial to be able to formally analyze the behavior of these processes, especially in safety-critical applications. Most of the existing formal methods are not designed for continuous-time processes, and those that are typically suffering from state explosion in practice. This article introduces a theoretical framework and a scalable computational method for formal analysis and control synthesis for switched diffusions, a class of stochastic models with linear dynamics that are continuous in both time and space domains; the focus is on safety with possible extensions to other properties. The proposed framework first constructs a finite abstraction in the form of an uncertain Markov process through discretization of both time and space domains. The errors caused by the discretization in each domain are formally characterized and cast into the abstraction model. Then, a strategy that maximizes the probability of the safety property and is robust against the errors is synthesized over the abstraction model. Finally, this robust strategy is mapped to a switching strategy for the stochastic processes that guarantees the safety property. The framework is demonstrated in three case studies, including one that illustrates the tradeoff of the error contribution by the time and space discretization parameters.