Turnpike in infinite dimension

TL;DR

This paper extends the turnpike theory to infinite-dimensional spaces, showing that under certain conditions, optimal feasible paths converge to a unique maximizer, with new cases included beyond previous literature.

Contribution

It introduces a novel turnpike result in infinite-dimensional spaces, establishing convergence of optimal paths under broad conditions and providing new illustrative examples.

Findings

Feasible paths converge to the unique maximizer under certain conditions.

The results include new cases not previously addressed in the literature.

The paper provides examples justifying the hypotheses and illustrating the scope.

Abstract

Let $\Phi$ be a correspondence from a normed vector space $X$ into itself, let $u: X\to \mathbf{R}$ be a function, and $\mathcal{I}$ be an ideal on $\mathbf{N}$. Also, assume that the restriction of $u$ on the fixed points of $\Phi$ has a unique maximizer $\eta^\star$. Then, we consider feasible paths $(x_0,x_1,\ldots)$ with values in $X$ such that $x_{n+1} \in \Phi(x_n)$ for all $n\ge 0$. Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots)$ which maximizes the smallest $\mathcal{I}$-cluster point of the sequence $(u(x_0),u(x_1),\ldots)$ is necessarily $\mathcal{I}$-convergent to $\eta^\star$. We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.