This paper is concerned with a compositional approach for constructing finite Markov decision processes of interconnected discrete-time stochastic control systems. The proposed approach leverages the interconnection topology and a notion of so-called stochastic storage functions describing joint dissipativitytype properties of subsystems and their abstractions. In the first part of the paper, we derive dissipativitytype compositional conditions for quantifying the error between the interconnection of stochastic control subsystems and that of their abstractions. In the second part of the paper, we propose an approach to construct finite Markov decision processes together with their corresponding stochastic storage functions for classes of discrete-time control systems satisfying some incremental passivablity property. Under this property, one can construct finite Markov decision processes by a suitable discretization of the input and state sets. Moreover, we show that for linear stochastic control systems, the aforementioned property can be readily checked by some matrix inequality. We apply our proposed results to the temperature regulation in a circular building by constructing compositionally a finite Markov decision process of a network containing 200 rooms in which the compositionality condition does not require any constraint on the number or gains of the subsystems. We employ the constructed finite Markov decision process as a substitute to synthesize policies regulating the temperature in each room for a bounded time horizon.

This paper is concerned with a compositional approach for constructing abstractions of interconnected discretetime stochastic control systems. The abstraction framework is based on new notions of so-called stochastic simulation functions, using which one can quantify the distance between original interconnected stochastic control systems and their abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results over concrete ones. In the first part of the paper, we derive sufficient small-gain type conditions for the compositional quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. In the second part of the paper, we focus on the class of discrete-time linear stochastic control systems with independent noises in the abstract and concrete subsystems. For this class of systems, we propose a computational scheme to construct abstractions together with their corresponding stochastic simulation functions. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 4 dimensions) of the interconnection of four discrete-time linear stochastic control subsystems (together 100 dimensions) in a compositional fashion.

This paper is concerned with a compositional approach for constructing infinite abstractions of interconnected discrete-time stochastic control systems. The proposed approach uses the interconnection matrix and joint dissipativity-type properties of subsystems and their abstractions described by a new notion of so-called stochastic storage functions. The interconnected abstraction framework is based on new notions of so-called stochastic simulation functions, constructed compositionally using stochastic storage functions of components. Using stochastic simulation functions, one can quantify the distance between original interconnected stochastic control systems and interconnected abstractions in the probabilistic setting. Accordingly, one can leverage the proposed results to perform analysis and synthesis over abstract interconnected systems, and then carry the results back over concrete ones. In the first part of the paper, we derive dissipativity-type compositional reasoning for the quantification of the distance in probability between the interconnection of stochastic control subsystems and that of their abstractions. Moreover, we focus on a class of discrete-time nonlinear stochastic control systems with independent noises in the abstract and concrete subsystems, and propose a computational scheme to construct abstractions together with their corresponding stochastic storage functions. In the second part of the paper, we consider specifications expressed as syntactically co-safe linear temporal logic formulae and show how a synthesized policy for the abstract system can be refined to a policy for the original system while providing guarantee on the probability of satisfaction. We demonstrate the effectiveness of the proposed results by constructing an abstraction (totally 3 dimensions) of the interconnection of three discrete-time nonlinear stochastic control subsystems (together 222 dimensions) in a compositional fashion such that the compositionality condition does not require any constraint on the number or gains of the subsystems. We also employ the abstraction as a substitute to synthesize a controller enforcing a syntactically co-safe linear temporal logic specification.

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