Turnpike property for extremals of variational problems with vector-valuedfunctions

In this paper we study the structure of extremals v: [0, T] --> R-n of vari ational problems with large enough T, fixed end points and an integrand f f rom a complete metric space of functions. We will establish the turnpike pr operty for a generic integrand f. Namely, we will show that for a generic i ntegrand f, any small epsilon > 0 and an extremal v: [0, T] --> R-n of the variational problem with large enough T, fixed end points and the integrand f, for each tau is an element of [L-1, T - L-1] the set (v(t): t is an ele ment of [tau, tau + L-2]) is equal to a set H(f) up to epsilon in the Hausd orff metric. Here H(f) subset of R-n is a compact set depending only on the integrand f and L-1 > L-2 > 0 are constants which depend only on epsilon a nd , .