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Evolutionary Systems and Control: Viability and Capture Analysis

Explore the relationship between evolutionary systems and control with a focus on viability and capture basins, using concepts like Viability Kernel and Marchaud Maps. Learn about the Single-Valued Regulation Maps and more.

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Evolutionary Systems and Control: Viability and Capture Analysis

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  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 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

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