Well posedness for differential inclusions on closed sets (Q807798)

Extending previous results, the authors prove a theorem stating the existence of a continuous selection \(y\mapsto x(.;y)\in AC([0,T];{\mathbb{R}}^ n),\) of solutions remaining on a closed subset \(K\subset {\mathbb{R}}^ n\), of a differential inclusion \(x'\in F(t,x)\), \(x(0)=y\in K\), in the case F(.,.) has closed nonempty values, is of Carathéodory-Lipschitz type and satisfies the Nagumo tangential condition: \(F(t,x)\subset T_ K(x)\forall (t,x)\in [0,T]\times K\), where \(T_ K(x)\) denotes the contingent cone to K at \(x\in K\). As corollaries of the theorem, the arcwise connectedness of the set of solutions and of the attainable sets is obtained.