A homology theory for hybrid systems: hybrid homology

Authors:
Aaron D. Ames;Shankar Sastry
Affiliations:
Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA;Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA
Venue:
HSCC'05 Proceedings of the 8th international conference on Hybrid Systems: computation and control
Year:
2005

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Abstract

By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space” of a hybrid system—its hybrid space or H-space. The homotopy colimit can be applied to this H-space to obtain a single topological space; the hybrid homology of an H-space is the homology of this space. The result is a spectral sequence converging to the hybrid homology of an H-space, providing a concrete way to compute this homology. Moreover, the hybrid homology of the H-space underlying a hybrid system gives useful information about the behavior of this system: the vanishing of the first hybrid homology of this H-space—when it is contractible and finite—implies that this hybrid system is not Zeno.