Necessary conditions for a nonlinear control system
The authors prove that if x(t) is a boundary trajectory of a Lipschitz continuous differential inclusion \(x\in F(t,x)\), \(x(0)=0\), the tangent cone to F(t,x(t)) at x(t) and that of the attainable set E(t) at x(t) coincide for almost every t provided that the boundary of F(t,x) is smooth. This is a noteworthy extension of \textit{H. Hermes} [J. Differ. Equations 3, 256-270 (1967; Zbl 0155.426)] and the first author [Astérisque 75-76, 187-197 (1980; Zbl 0459.49028)]. They also prove that the outward normal to these cones along the trajectory is Lipschitz continuous in t. Using the lower one-sided directional derivative the paper obtains first order necessary conditions which are stronger than existing ones. Two examples are given.
scientific article; zbMATH DE number 4046502