Saddle points of hamiltonian systems in convex problems of Lagrange

In Lagrange problems of the calculus of variations where the LagrangianL(x ), not necessarily differentiable, is convex jointly inx and , optimal arcs can be characterized in terms of a generalized Hamiltonian differential equation, where the HamiltonianH(x, p) is concave inx and convex inp. In this paper, the Hamiltonian system is studied in a neighborhood of a minimax saddle point ofH. It is shown under a strict concavity-convexity assumption onH that the point acts much like a saddle point in the sense of differential equations. At the same time, results are obtained for problems in which the Lagrange integral is minimized over an infinite interval. These results are motivated by questions in theoretical economics.This research was supported in part by Grant No. AFOSR-71-1994.