Turnpike Property for Extremals of Variational Problems with Vector-valued Functions

In this paper we study the structure of extremals ν : [0, T ] → Rn of variational problems with large enough T , fixed end points and an integrand f from a complete metric space of functions. We will establish the turnpike property for a generic integrand f . Namely, we will show that for a generic integrand f , any small ε > 0 and an extremal ν : [0, T ] → Rn of the variational problem with large enough T , fixed end points and the integrand f , for each τ ∈ [L1, T − L1] the set {ν(t) : t ∈ [τ, τ + L2]} is equal to a set H(f) up to ε in the Hausdorff metric. Here H(f) ⊂ Rn is a compact set depending only on the integrand f and L1 > L2 > 0 are constants which depend only on ε and |ν(0)|, |ν(T )|.