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A composition theorem for bisimulation functions

The standard engineering approach to modelling of complex systems is highly compositional. In order to be able to understand (or to control) the behavior of a complex dynamical systems, it is often desirable, if not necessary, to view this system as an interconnection of smaller interacting subsystems, each of these subsystems having its own functionalities. In this paper, we propose a compositional approach to the computation of bisimulation functions for dynamical systems. Bisimulation functions are quantitative generalizations of the classical bisimulation relations. They have been shown useful for simulation-based verification or for the computation of approximate symbolic abstractions of dynamical systems. In this technical note, we present a constructive result for the composition of bisimulation functions. For a complex dynamical system consisting of several interconnected subsystems, it allows us to compute a bisimulation function from the knowledge of a bisimulation function for each of the subsystem.