A Homology Theory for Hybrid Systems: Hybrid Homology - CiteSeerX

A Homology Theory for Hybrid Systems: Hybrid Homology 101Proposition 1. Let H =(H, S) be the finite H-space obtained from the classicalH-space H class =(Γ, D, G, R). IfH is contractible, thenand it follows that χ(H) =χ(Γ ).HH n (H, R) ∼ = H n (Γ, R)A rather startling point is that the underlying H-space of a hybrid system—more specifically its homology—in some way dictates the behavior that thishybrid system can display (for a complete discussion on this, as well examplesand a review of Zeno behavior, see [4]). Even more importantly, the type ofbehavior that the homology of an H-space “notices” is exactly the behavior thatis central, and unique, to hybrid systems: Zeno behavior. This point is mademore clear in the following theorem:Theorem 8. Let H H =(H, S H ):=(H, P Top ◦ S H ) be the underlying H-spaceof the hybrid system H =(H, S H ).IfH H is contractible and finite, thendim R HH 1 (H H , R) = dim R N (K U(H) )=0 ⇒ H is not Zeno.If H H is connected, it implies that dim R HH 0 (H H , R) = 1, and so we havethe following corollary to this theorem which is in a form more reminiscent of“Morse-type” theorems.Corollary 6. If H His connected, contractible and finite, thenχ(H H )=|Ob(H)|−|Mor id/ (H)| =1 ⇒ H is not Zeno.In many ways, this theorem (and its corollary) is more of a “Morse-type”theorem than Theorem 4. The hope is, through the use of the categorical frameworkfor hybrid systems introduced here, to incorporate the dynamics of a hybridsystem into the above theorems in order to obtain tighter algebraic theorems onthe nonexistence of Zeno.Example 4. For the water tank hybrid space H W , using Proposition 1, it is easyto see that HH 1 (H W , R) ∼ = HH 0 (H W , R) ∼ = R. So we cannot say that the watertank is not Zeno, which is good because it is Zeno.References1. Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations.Volume 304 of Lecture Notes in Mathematics. Springer-Verlag (1972)2. Thomason, R.W.: First quadrant spectral sequences in algebraic K-theory. InDupont, J.L., Madsen, I.H., eds.: Algebraic Topology. Volume 763 of Lecture Notesin Mathematics. Springer-Verlag (1978) 332–3553. Vogt, R.M.: Homotopy limits and colimits. Mathematische Zeitschrift 134 (1973)11–524. Ames, A.D., Sastry, S.: Characterization of Zeno behavior in hybrid systems usinghomological methods. Submitted to ACC (2005)