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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
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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
Set valued formulation of controlled systems is equivalent to where
Evolutionary systems associated with control systems [Aubin, Notes de cours, ENS Cachan, 2002]
Evolutionary systems (continued) [ABBSP, 2007]
Viability and capturability [ABBSP, 2007]
Viability kernel (I) For the rest of this study (unless stated otherwise), we will consider the following differential inclusion, referred to as (0.1)
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]
Capture basin [ABBSP, 2007]
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.
Example: the Zermelo swimmer (I) [Saint-Pierre, 1997, 2006]
Example: the Zermelo swimmer (II) [Saint-Pierre, 1997, 2006]
Example: the Zermelo swimmer (III) [Saint-Pierre, 1997, 2006]
Viability kernel with target, capture basin [ABBSP, 2008]
Regulation maps Single valued [selection] from the regulation map: feedback