Lagrange-type extremal trajectories in differential inclusions
The authors consider the differential inclusion \((*)\) \(x'\in F(t,x)\), where the multifunction \(F\) is defined for \(t\in I=[0,1]\) and \(x\in \Omega\) when \(\Omega\) is an open subset of \(\mathbb{R}^ n\). Denote by \(B(F)\) the set of all boundary trajectories of the inclusion \((*)\). Let also \(H(F)\) the family of all Hamiltonian extremals of the inclusion \((*)\), \(L(F)\) the family of all Lagrangian extremals of the inclusion \((*)\) and \(L_{co}(F)\) the class of all relaxed Lagrangian extremals of \((*)\). The authors give in the first part of the paper a comparison of the above three classes of extremals \(H(F)\), \(L(F)\) and \(L_{co}(F)\) presenting seven examples of autonomous, two-dimensional versions of differential inclusion \((*)\). Another approach of comparing types of extremality conditions was given by \textit{P. D. Loewen} and \textit{R. T. Rockafellar} [Trans. Am. Math. Soc. 325, No. 1, 39-72 (1991; Zbl 0734.49009)]. In the second part they prove, without any convexity hypotheses, that every boundary trajectory is a Lagrangian extremal, that is \(B(F)\subset L(F)\).
scientific article; zbMATH DE number 86072
scientific article