Saddle Points of Hamiltonian Systems in Convex Problems of Lagrange

In Lagrange problems of the calculus of variations where the Lagrangian L(x, (x) over dot), not necessarily differentiable, is convex jointly in x and (x) over dot, optimal arcs can be characterized in terms of a generalized Hamiltonian differential equation, where the Hamiltonian H( x, p) is concave in x and convex in p. In this paper, the Hamiltonian system is studied in a neighborhood of a minimax saddle point of H. It is shown under a strict concavity-convexity assumption on H 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.