Conley's fundamental theorem for a class of hybrid systems

Kvalheim, Matthew D., Paul Gustafson, and Daniel E. Koditschek. "Conley's fundamental theorem for a class of hybrid systems." arXiv preprint arXiv:2005.03217 (2020).

The law of conservation of energy states that energy is neither created nor destroyed: the total energy of an isolated physical system remains constant over time. But for a non-isolated physical system, the total energy typically decreases until the system comes to rest at some steady state (unless energy is pumped into the system). For example, friction will eventually bring a swinging pendulum to a halt (unless it receives another push). Viewed more abstractly, such a physical system can be understood as moving downhill along an energy landscape until it reaches the lowest point of the "valley" to which the initial physical state belongs.

Mathematical models of physical systems such as those governed by Newton's laws are examples of the more general dynamical systems which mathematically model processes that change over time. Amazingly, in the late 1970s the mathematician C. Conley showed that all "reasonable" dynamical systems have surprisingly much in common with the physical systems mentioned above: they all have "generalized energy landscapes", and any initial state of the system evolves by moving downhill along the landscape until reaching the bottom of the valley to which it belongs. This powerful insight, established for dynamical systems evolving continuously in time, was also shown to apply to dynamical systems evolving at discrete time instants by the mathematician J. Franks in the late 1980s.

However, many physical systems—such as robots, cyber-physical systems, and pinball machines—are best modeled by dynamical systems which are hybrid in the sense that they involve both continuous-time evolution and discrete-time "events". While important contributions have been made by many researchers, the theory of such "hybrid systems" is still in its infancy, and it was unknown whether the “generalized energy landscape” insights of Conley and Franks might extend to hybrid systems. In a recent arXiv preprint [KGK20] M. D. Kvalheim, P. Gustafson, and D. E. Koditschek have shown that the insights of Conley and Franks do indeed extend to a broad class of hybrid systems relevant, e.g., for robotics. The proof of this fact in [KGK20] works by "stretching" and "gluing" the "space of all states" of the hybrid system to produce a certain continuous-time dynamical system to which Conley’s results can be applied and subsequently pulled back to the hybrid system. A more technical description of this work is given in the abstract here.

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