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 … Show more
“…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.…”
Section: Discussionmentioning
confidence: 99%
“…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.…”
Section: A Hybrid Systemsmentioning
confidence: 99%
“…1). In the present paper we will not need a formal definition of "execution"; see [38,Def. 2.3] for one example.…”
Section: A Hybrid Systemsmentioning
confidence: 99%
“…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]).…”
Section: B Motivation: Stationary Point Separation Principlementioning
A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained. For the hybrid systems considered, guard sets are not assumed to be smooth; distinct "modes" are not assumed to have constant dimension; and resets are arbitrary multivalued maps (relations).
“…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.…”
Section: Discussionmentioning
confidence: 99%
“…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.…”
Section: A Hybrid Systemsmentioning
confidence: 99%
“…1). In the present paper we will not need a formal definition of "execution"; see [38,Def. 2.3] for one example.…”
Section: A Hybrid Systemsmentioning
confidence: 99%
“…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]).…”
Section: B Motivation: Stationary Point Separation Principlementioning
A generalization of the Poincaré-Hopf index theorem applicable to hybrid dynamical systems is obtained. For the hybrid systems considered, guard sets are not assumed to be smooth; distinct "modes" are not assumed to have constant dimension; and resets are arbitrary multivalued maps (relations).
In this chapter we discuss differentiable structures on topological manifolds. In particular, we discuss transversality, tubular neighbourhoods, index theory, the degree and the theorems by Poincaré and Hopf and Bobylev and Krasnosel’skiĭ.
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