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The Turnpike property and the long-time behavior of the Hamilton-Jacobi equation

Submission Status:under review
  • In this work, we analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon TTT tends to infinity, the value function asymptotically behaves as W(x)+c T+λW(x) + c\, T + \lambda W(x)+cT+λ, and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we provide a new turnpike result for the linear-quadratic optimal control problem with constrained control. As a main feature, our turnpike result applies to the case when the steady optimum may saturate the control constraints. This prevented us from proving the turnpike property with an exponential rate, which is well-known to hold for the unconstrained case.

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Metadaten
Author:C Esteve, H Kouhkouh, D Pighin, E Zuazua
Document Type:Preprint
Language:English
Date of Publication (online):2021/04/23
Date of first Publication:2021/10/08
Release Date:2021/10/08
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Subprojects:C03
C05
Licence (German):License LogoCreative Commons - CC BY-SA - Namensnennung - Weitergabe unter gleichen Bedingungen 4.0 International