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On the Stability of Zeno Equilibria

Aaron D. Ames1, Paulo Tabuada2, and Shankar Sastry1

1 Department of Electrical Engineering and Computer Sciences,University of California at Berkeley, Berkeley, CA 94720

{adames, sastry}@eecs.berkeley.edu2 Department of Electrical Engineering,

University of Notre Dame, Notre Dame, IN 46556ptabuada@nd.edu

Abstract. Zeno behaviors are one of the (perhaps unintended) featuresof many hybrid models of physical systems. They have no counterpartin traditional dynamical systems or automata theory and yet they haveremained relatively unexplored over the years. In this paper we addressthe stability properties of a class of Zeno equilibria, and we introducea necessary paradigm shift in the study of hybrid stability. Motivatedby the peculiarities of Zeno equilibria, we consider a form of asymptoticstability that is global in the continuous state, but local in the discretestate. We provide sufficient conditions for stability of these equilibria,resulting in sufficient conditions for the existence of Zeno behavior.

1 Introduction

Hybrid models have been used successfully during the past decade to describesystems exhibiting both discrete and continuous dynamics, while they have si-multaneously allowed complex models of continuous systems to be simplified.We are interested in the rich dynamical behavior of hybrid models of physi-cal systems. These hybrid models admit a kind of equilibria that is not found incontinuous dynamical systems or in automata theory: Zeno equilibria. Zeno equi-libria are collections of points which are invariant under the discrete componentof the hybrid dynamics, and which can be stable is many cases of interest.

Mechanical systems undergoing impacts are naturally modeled as hybrid sys-tems (cf. [1] and [2]). The convergent behavior of these systems is often ofinteresteven if this convergence is not to classical notions of equilibriumpoints. This motivates the study of Zeno equilibria because even if the conver-gence is not classical, it still is important. For example, simulating trajectoriesof these systems is an important component in their analysis, yet this may notbe possible due to the relationship between Zeno equilibria and Zeno behavior.

An equally important reason to address the stability of Zeno equilibria is tobe able to assess the existence of Zeno trajectories. This behavior is infamous inthe hybrid system community for its ability to halt simulations. The only way to

This research is supported by the National Science Foundation (award numbersCCR-0225610 and CSR-EHS-0509313).

J Hespanha and A. Tiwari (Eds.): HSCC 2006, LNCS 3927, pp. 3448, 2006.c Springer-Verlag Berlin Heidelberg 2006.

On the Stability of Zeno Equilibria 35

prevent this undesirable outcome is to give a priori conditions on the existenceof Zeno behavior. This has motivated a profuse study of Zeno hybrid systems(see [1, 3, 4, 5, 6, 7, 8] to name a few) but a concrete notion of convergence (in thesense of stability) has not yet been introduced. As a result, there is a noticeablelack of sufficient conditions for the existence of Zeno behavior. We refer thereader to [3, 7, 8] for a more thorough introduction to Zeno behavior.

Our investigations into the stability of Zeno equilibria are made possiblethrough a categorical framework for hybrid systems (as first introduced in [9]and later utilized in [10]). This theory allows non-hybrid objects to be gener-alized to a hybrid setting. Specifically, let T be a category, i.e., a collection ofmathematical objects that share a certain property together with morphisms be-tween these objects. A hybrid object over this category is a special type of smallcategory H, termed an H-category, together with a functor (either covariant orcontravariant) S : H T . Morphisms between objects of T are generalized to ahybrid setting through the use of natural transformations.

The main contribution of this paper is sufficient conditions for the stabilityof Zeno equilibria. As a byproduct, we are able to give sufficient conditionsfor the existence of Zeno behavior. The categorical approach to hybrid systemsallows us to decompose the study of stability into two manageable steps. Thefirst step consists of identifying a sufficiently rich, yet sufficiently simple, class ofhybrid systems embodying the desired stability properties: first quadrant hybridsystems. The second step is to understand the stability of general hybrid systemsby understanding the relationships between these systems and first quadranthybrid systems described by morphisms (in the category of hybrid systems).

2 Classical Stability: A Categorical Approach

In this section we revisit classical stability theory under a categorical light. Thenew perspective afforded by category theory is more than a simple exercise inabstract nonsense because it motivates the development of an analogous stabilitytheory for hybrid systems and hybrid equilibria to be presented in Sections 4and 5. We shall work on Dyn, the category of dynamical systems, which has asobjects pairs (M, X), where M is a smooth manifold1 and X : M TM is asmooth vector field. The morphisms are smooth maps f : N M making thefollowing diagram commutative:

TNTf TM

N

Y

f M

X

(1)

The subcategory Interval(Dyn) of Dyn will play an especially important role in thetheory developed in this paper. This subcategory is the full subcategory of Dyn1 We assume that M is a Riemannian manifold, and so has a metric d(x, y) = xy.

Alternatively, we could assume that M is a subset of Rn.

36 A.D. Ames, P. Tabuada, and S. Sastry

defined by objects2 (I, ddt) with I a subset of R of the form [t, t], (t, t], [t, t), (t, t)

and {t}, where [t, t] is a manifold with boundary (and so is (t, t] and [t, t)) and{t} is a zero-dimensional manifold consisting of the single point t (which istrivially a smooth manifold). The following observation shows the relevance ofInterval(Dyn). A morphism c : (I, d/dt) (M, X) is a smooth map c : I Mmaking diagram (1) commutative and thus satisfying:

c(t) = Tc ddt

= X c(t).

We can therefore identify a morphism c : (I, d/dt) (M, X) with a trajectory of(M, X). Furthermore, the existence of a morphism f : (N, Y ) (M, X) impliesthat for every trajectory c : (I, d/dt) (N, Y ), the composite f c : (I, d/dt) (M, X) is a trajectory of (M, X). In other words, a morphism f : (N, Y ) (M, X) carries trajectories of (N, Y ) into trajectories of (M, X).

Remarkably, stability also can be described through the existence of certainmorphisms. Let us first recall the definition of globally asymptotically stableequilibria.

Definition 1. Let (M, X) be an object of Dyn. An equilibrium point x Mof X is said to be globally asymptotically stable when for any morphism c :([t, ), ddt ) (M, X), for any t1 > t and for any > 0 there exists a > 0satisfying:

1. c(t1) x < c(t2) x < t2 t1 t,2. lim c() = x.

Consider now the full subcategory of Dyn denoted by GasDyn and defined byobjects (R+0 , ) where is a class K function. Lyapunovs second methodcan then be described as follows:

Theorem 1. Let (M, X) be an object of Dyn. An equilibrium point x M ofX is globally asymptotically stable if there exists a morphism:

(M, X)v (R+0 , ) GasDyn

in Dyn satisfying:

1. v(x) = 0 implies x = x,2. v : M R+0 is a proper (radially unbounded) function.

The previous result suggests that the study of stability properties can be car-ried out in two steps. In the first step we identify a suitable subcategory havingthe desired stability properties. In the case of global asymptotic stability, thissubcategory is GasDyn; for local stability we could consider the full subcategorydefined by objects of the form (R+0 , ) with a non-negative definite function.2 We do not consider more general objects of the form (J, g(t)d/dt) with g > 0 since

each such object is isomorphic to (I, d/dt).

On the Stability of Zeno Equilibria 37

The chosen category corresponds in some sense to the simplest possible objectshaving the desired stability properties. In the second step we show that existenceof a morphism from a general object (M, X) to an object in the chosen subcat-egory implies that the desired stability properties also hold in (M, X). This isprecisely the approach we will develop in Sections 4 and 5 for the study of Zenoequilibria.

3 Categorical Hybrid Systems

This section is devoted to the study of first quadrant hybrid systems, categoricalhybrid systems, and their interplay. We begin by defining a very simple class ofhybrid systems; these systems are easy to understand and analyze, but lack gen-erality. We then proceed to define general hybrid systems through the frameworkof hybrid category theory; these systems are general but difficult to analyze. Theadvantage of introducing these two concepts is that not only can they be relatedthrough explicit constructions, but also through the more general framework ofmorphisms in the category of hybrid systems. This relationship will be importantin understanding the stability of general hybrid systems.

First Quadrant Systems. In order to understand the stability of generalhybrid systems, we must consider a class of hybrid systems analogous to theobjects of GasDyn; these are termed first quadrant hybrid systems. It is notsurprising that these would be chosen as the canonical hybrid systems withwhich to understand the stability of Zeno equilibria as they already have beenused to derive sufficient conditions for the existence of Zeno behavior in [3].

A first quadrant hybrid system is a tuple:

HFQ = (, D, G, R, F ),

where

= (Q, E) is an orien