Turnpike in infinite dimension

Leonetti, Paolo ; Caprio, Michele

Canadian mathematical bulletin, Tome 65 (2022) no. 2, pp. 416-430

Résumé

Let $\Phi $ be a correspondence from a normed vector space X into itself, let $u: X\to \mathbf {R}$ be a function, and let $\mathcal {I}$ be an ideal on $\mathbf {N}$. In addition, 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.

Détail

DOI : 10.4153/S0008439521000382

Mots-clés : Turnpike, ideal and statistical convergence, ideal cluster point, optimal stationary point, fixed point of correspondences