Abstract-The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunov-like inequalities, which for deterministic systems can be computed using recent sum-of-squares techniques. Our framework is illustrated on a safety verification example. The established notions for both discrete and continuous systems are all exact, requiring external behavior of two systems to be identical. When interacting with the physical world, exact equivalence notions are restrictive and not robust. Notions of system approximation seem much more appropriate in this context, and this has recently resulted in approximate bisimulation relations for stochastic [4] , quantitative [3] , and metric transition systems [7] , [8] . In [8], we developed a framework for (discrete and continuous) system approximation for general metric transition systems. The approximation process is based on a metric on the set of observations. Rather than requiring that the distance between system observations is (and remains) zero, we require that the distance between observations is (and remains) arbitrarily close. We showed that approximate bisimulations can be characterized by a set of functions called bisimulation functions. In [7] , a method to compute bisimulations functions is proposed for the class metric transition systems generated by linear systems with constrained inputs. In this paper, we extend our work to the class of nonlinear dynamical systems 1 . We develop Lyapunov-like
Especially related to the work described in this paper is the recent notion of approximate bisimulation introduced in REF .
Approximate Bisimulations for Nonlinear Dynamical Systems
Proceedings of the 44th IEEE Conference on Decision and Control
Proceedings of the 44th IEEE Conference on Decision and Control
Computer Science