Turnpike property for extremals of variational problems with vector-valued functions

In this paper we study the structure of extremals $\nu\colon[0,T]\to R^n$ 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 $\varepsilon>0$ and an extremal $\nu\colon[0,T]\to R^n$ of the variational problem with large enough $T$, fixed end points and the integrand $f$, for each $\tau\in[L_1, T-L_1]$ the set $\{\nu(t)\colon t\in[\tau,\tau+L_2]\}$ is equal to a set $H(f)$ up to $\varepsilon$ in the Hausdorff metric. Here $H(f)\subset R^n$ is a compact set depending only on the integrand $f$ and $L_1>L_2>0$ are constants which depend only on $\varepsilon$ and $|\nu(0)|$, $|\nu(T)|$.