Controllability and extremality in nonconvex differential inclusions
Let \(F: [0, T]\times \mathbb{R}^ n\to 2^{\mathbb{R}^ n}\) be a set-valued map with compact values; let \(\eta: \mathbb{R}^ n\to \mathbb{R}^ m\) be a locally Lipschitzian map, \(z(t)\) a given trajectory, and \(\mathcal R\) the reachable set at \(T\) of the differential inclusion \(\dot x(t)\in F(t, x(t))\). We prove sufficient conditions for \(\eta(z(T))\in \text{int }\eta({\mathcal R})\) and establish necessary conditions in maximum principle form for \(\eta(z(T))\in \partial\eta({\mathcal R})\). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.
scientific article; zbMATH DE number 790575
scientific article