Conley's Fundamental Theorem for a Class of Hybrid Systems

Abstract

We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a globally defined hybrid complete Lyapunov function exists for every hybrid system in this class. Motivated by mechanics and control settings where physical or engineered events cause abrupt changes in a system's governing dynamics, our results apply to a large class of Lagrangian hybrid systems (with impacts) studied extensively in the robotics literature. Viewed formally, these results generalize those of Conley and Franks for continuous-time and discrete-time dynamical systems, respectively, on metric spaces. However, we furnish specific examples illustrating how our statement of sufficient conditions represents merely an early step in the longer project of establishing which formal assumptions can and cannot endow hybrid systems models with the topologically well-characterized partitions of limit behavior that make Conley's theory so valuable in those classical settings.

Poincaré-Hopf theorem for hybrid systems

“…A different generalization of the Poincaré-Hopf theorem not mentioned in §I is due to McCord [54]; this result says that the sum of the Hopf indices of those zeros of a smooth vector field contained in an isolated invariant set S is equal to the Euler characteristic of the homology Conley index of S. It would be interesting to develop a Conley index theory [55], [56], [57] for hybrid systems and to investigate whether an analogous relationship holds in the hybrid setting. A Conley index theory for hybrid systems would also be interesting since, together with the results of [38], it would constitute a first theoretical step toward extending rigorous computerassisted analysis methods based on Conley theory [58], [59] to hybrid dynamical systems.…”

“…The results of this paper apply to a broad subclass of such systems. The notation just introduced follows that of [38], but the setup just described is (mostly) more general. General hybrid systems have both discretetime and continuous-time behavior; this is captured in typical definitions of executions of hybrid systems (Fig.…”

“…1). In the present paper we will not need a formal definition of "execution"; see [38,Def. 2.3] for one example.…”

“…One might expect the structure of all H-stationary points to be strongly constrained by the topology of spaces associated to H such as the hybrifold [10], [13] or hybrid suspension [38,Fig. 6] (or "homotopy colimit" [34], or "1relaxed hybrid quotient space" [39]).…”