A uniqueness theorem for differential inclusions
The author considers the autonomous differential inclusion problem \(\dot x\in F(x(t))\) a.e. \(t\in [0,T]\), \(x(0)=\xi\), where x: [0,T]\(\to {\mathbb{R}}^ n\) is an absolutely continuous vector function, F a multifunction assigning to each point in an open subset X of \({\mathbb{R}}^ n\) a nonempty compact subset of \({\mathbb{R}}^ n\), and \(\xi\) a given vector in \({\mathbb{R}}^ n\). By the reachable set \(R^{(T)}(\xi)\) is understood the set of vectors x(T) for x(t) being a solution of the above problem. He observes that the family \(\{R^ T\}_{T\geq 0}\) of multifunctions is a semigroup since \(R^{(s+t)}(\xi)=\cup_{\eta \in R^{(t)}(\xi)}R^{(s)}\eta =:(R^ s\circ R^{(t)})\xi\) for s,t\(\geq 0\) and that the Hausdorff distance between the set \((R^{(t)}(\xi)-\xi)/t\) and F(\(\xi\)) tends to zero as \(t\to 0\). Thus, the semigroup \(R^{(t)}\) possesses an infinitesimal generator. These facts enable him to show that if F is locally Lipschitzian on X then the reachable sets \(R^{(t)}(\xi)\) exhibit the same uniqueness properties as the solutions of initial value problems of ordinary differential equations.
scientific article; zbMATH DE number 4156936
scientific article