A Homology Theory for Hybrid Systems: Hybrid Homology - CiteSeerX

92 A.D. Ames and S. SastryMf ✲ NV❄ F✲TNW❄TMcommutes, and for each p ∈ M the restriction of F to the fiber T p M, F | TpM :T p M → T f(p) N, is linear. In the case when F is the pushforward of f, i.e.,F = f ∗ , this definition implies that V and W are f-related (cf. [7]). A smoothdynamical subsystem is a pair (S ⊆ M,V | S ) where S is an embedded submanifoldof M and V | S is the restriction of a vector field V on M to S, andhence a vector field along S. As in the case of dynamical systems, morphisms ofsmooth dynamical subsystems are given in a way analogous to the definition ofmorphisms of dynamical systems (cf. [8] for a definition).Note that there is a projection functor P Top : Dyn → Top from the categoryof dynamical systems to the category of topological spaces given by P Top (X, ϕ) =X on objects and P Top (h, r) =h on morphisms. Similarly, we have a projectionfunctor from the category Sdyn to the category Man, P Man : Sdyn → Mandefined in an analogous way.Hybrid Systems. With the definitions of dynamical systems and smooth dynamicalsystems in hand, we can define hybrid systems. A classical hybrid systemis a tuple H class =(H class ,Φ)=(Q, E, D, G, R, Φ) where H class is a classical H-space and Φ = {ϕ i } i∈Q where ϕ i is a flow on the topological space D i , i.e.,(D i ,ϕ i ) is a dynamical system for each i ∈ Q.Similarly, we can define smooth classical hybrid systems as pairs G class =(G class ,V)=(Q, E, M, G M ,R S ,V) where G class is a classical G-space and V ={V i } i∈Q where V i is a smooth vector field on the manifold M i , i.e., (M i ,V i )isasmooth dynamical system for each i ∈ Q.Definition 2. A categorical hybrid system is a pair H cat = (H, S H ) whereH is an H-small category and S H : H → Dyn is a functor such that the pair(H, P Top ◦ S H ) is a categorical H-space. The H-spaceH H =(H, P Top ◦ S H ):=(H, S H H )is referred to as the underlying H-space of the hybrid system H cat .Theorem 2. If for each e ∈ E there exists a morphism of dynamical sub-systemsα e :(G e ⊆ D s(e) ,ϕ s(e) | Ge ) → (R e (G e ) ⊆ D t(e) ,ϕ t(e) | Re(G e)),then there is an injective correspondence{Classical Hybrid Systems, H class }−→{Categorical Hybrid Systems, H cat }.