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

A. J. Zaslavski
1999 Transactions of the American Mathematical Society  
In this paper we study the structure of extremals ν : [0, T ] → 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 ε > 0 and an extremal ν : [0, T ] → R n of the variational problem with large enough T , fixed end points and the integrand f , for each τ ∈ [L 1 , T − L 1 ] the set {ν(t): t
more » ... ∈ [τ, τ + L 2 ]} is equal to a set H(f ) up to ε in the Hausdorff metric. Here H(f ) ⊂ R n is a compact set depending only on the integrand f and L 1 > L 2 > 0 are constants which depend only on ε and |ν(0)|, |ν(T )|.
doi:10.1090/s0002-9947-99-02132-7 fatcat:shtilqz4jjf6bgnn7uaj75hqxe