1. Evolutionary systems, evolutions
Marchaud maps, Lipschitz maps, Filippov’s theorem
Set valued representation of control systems
Viability kernel, viability kernel with target
Capture basin
Invariance kernel
Absorpsion basin
Regulation maps, viable and capturing evolutions
Tangential and normal characterization of viability kernels and capture basins

2. Marchaud maps

3. Lipschitz maps

4. Set valued formulation of controlled systems
is equivalent to
where

5. Evolutionary systems associated with control systems
[Aubin, Notes de cours, ENS Cachan, 2002]

6. Differential inclusions

7. Evolutionary systems (continued)
[ABBSP, 2007]

8. Viability and capturability
[ABBSP, 2007]

9. Viability kernel (I)
For the rest of this study (unless stated otherwise), we will consider the following differential inclusion, referred to as (0.1)

10. Viability kernel (II)

11. Viability kernel (III)

12. Example (environmental engineering): pollution-tax
pollution not acceptable
for x  2
economy not viable
for 0.2  x
I will first present a new model which I generated to do these tasks.
I will then present algorithms to treat the functionality separation problem.
I will briefly talk about safety assurance and finally about research goals.
pollution tax p should be positive
[Saint-Pierre, 1994, 1998]

13. Capture basin
[ABBSP, 2007]

14. Capture basin

15. Viability kernel with target
The viability kernel with target is the set of points from which at least one evolution stays in K forever or reaches C while staying in K.

16. Example: the Zermelo swimmer (I)
[Saint-Pierre, 1997, 2006]

17. Example: the Zermelo swimmer (II)
[Saint-Pierre, 1997, 2006]

18. Example: the Zermelo swimmer (III)
[Saint-Pierre, 1997, 2006]

19. Viability kernel with target, capture basin
[ABBSP, 2008]

20. Viability kernel with target, capture basin

21. Invariance kernel

22. Viability kernel and capture basin

23. Absorption basin

24. Absorption basin

25. Regulation maps
Single valued [selection] from the regulation map: feedback

26. Viable and capturing evolutions