1. Bisimulation Relation A lecture over E. Hagherdi, P. Tabuada, G. J. Pappas Bisimulation relation for dynamical, control, and hybrid systems Rafael Wisniewski Aalborg University Ph.D. course November 2005

2. Please ask as much as possible. I would be happy for all relevant to the topic questions.

3. Labeled Transition Systems

4. Product and Pullback Product of C 1 and C 2 Pullback

5. Product of Transition Systems

6. Strong Bisimulation Whenever commutes then commutes Open Maps:

7. Bran L Open Maps Bran L is a full subcategory of T L of all synchrinization trees with a single finite branch. P-bisimilarity:

8. Generalization of P-open maps We generalize P-open maps to the category Dyn of dynamical systems and Hyb the category of hybrid dynamical systems. The path category P as the full subcategory of Dyn with objects P : I → TI, where P(t) = (t, 1) and I is an open interval of R containing the origin. Morphism:

9. P-open Maps

10. P-bisimilarity for dynamical systems Pullback in the category of P-open surjective submersions:

11. Bisimilarity of Dynamical Systems

12. Example R Consider the vector field X on M = R 2 defined R Also consider the vector field Y on N = R defined by is a Dyn-morphism Then

13. Hybrid Dynamical Systems

14. Category Hyb Recall a time transition system from Henzinger The state space is Transition relation like in Henzinger

15. Path Category in Hyb The path category P is the full subcategory of Hyb: t0t0 t1t1 t2t2 t k-1 tktk dx/dt = 1

16. Example of a path Consider a path This path is represented by the path object P which has states l 0, l 1, l 2

17. P-open Maps for Hyb

18. Characterization of bisimulation in Hyb is said to be a bisimulation relation iff for all implies ▒▒

19. Bisimulation Characterization

20. Future Work Extension of the bisimulation notion from the article from timed transition systems to time abstract transition systems. This can be done by identify a whole flow line with a point in the space of flow lines. The strong simulation is too strong equivalence relation on dynamical systems is too strong. Try to use weaker equivalence relation some form of topological equivalency. On Friday 18th Nov. try to understand the definitions and go through proofs in the section dealing with the dynamical systems. If you understand P-open maps and bisimulation in the category of dynamical systems the generalization to hybrid systems seems natural.