Viable solutions to nonautonomous inclusions without convexity (Q1429402)

The authors consider the differential inclusion \(x'(t)\in F(t,x(t))\) a.e. for \(t\in [0,T]\) in a real Hilbert space \(X\), where \(F:[0,+\infty)\times K\to X\) is measurable in \(t\) and upper semicontinuous in \(x\), with \(K\) convex and locally compact. They prove that if there exists a lower regular function \(V:X\to \mathbb{R}\) such that \(T_K(x)\cap F(t,x)\cap\partial V(x)\neq \emptyset\) for every \(x\in K\) and a.e. for \(t\geq 0\) then, for each \(x_0\in K\), there exists \(T>0\) such that the differential inclusion has at least one solution \(x:[0,T]\to K\). Here \(T_K(x)\) is the Bouligand tangent cone to \(K\) at \(x\) and \(\partial V(x)\) is the Clarke subdifferential of \(V\) at \(x\).