Viability Theory and Regulation of Controled Systems: The Convex Case

Abstract

When we assume that the viability subset K is convex and compact, we obtain many more properties. The most striking one is that the tangential condition

$$\forall x \in K,\,F\left( x \right) \cap {T_K}\left( x \right) \ne \phi $$

((1))

which is necessary and sufficient when F has convex values for the differential inclusion

$$\begin{array}{*{20}{c}} {i)x\prime \left( t \right) \in F\left( {x\left( t \right)} \right),} \\ {ii)x\left( 0 \right) = {x_0},{x_0}{\text{given in K}},} \end{array}{\text{ }}$$

((2))

to have viable trajectories for all initial states x₀ in K, is also a sufficient condition for F to have an equilibrium state x̄ in K.