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

Marchaud maps

Lipschitz maps

Set valued formulation of controlled systems is equivalent to where

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

Differential inclusions

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)

Viability kernel (II)

Viability kernel (III)

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]

Capture basin [ABBSP, 2007]

Capture basin

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]

Viability kernel with target, capture basin

Invariance kernel

Viability kernel and capture basin

Absorption basin

Absorption basin

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

Viable and capturing evolutions