Turnpike Theorem for Convex Infinite Dimensional Discrete-Time Control Systems

In this work we study the structure of "approximate" solutions for an infinite dimensional discrete-time optimal control problem determined by a convex function v : K × K → $R^1$, where K is a convex closed bounded subset of a Banach space. We show that for a generic function v there exists $y_v$ ∈ K such that each "approximate" optimal solution $\{x_i\}_{i=0}^n$ ⊂ K is a contained in a small neighborhood of $y_v$ for all i ∈ {N,...,n - N}, where N is a constant which depends on the neighborhood and does not depend on n.