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

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  • 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 an optimal control problem. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation. In order to carry out our study, we use the setting of a finite-dimensional linear-quadratic optimal control problem, for which the turnpike property is well understood. We prove that, when the time horizon T tends to infinity, the value function converges to a travelling-front like solution of the form W(x) + c T + λ. In addition, we provide a control interpretation of each of these three terms in the spirit of the turnpike theory. Finally, we compare this asymptotic decomposition with the existing results on long-time behavior for Hamilton-Jacobi equations. We stress that in our case, the Hamiltonian is not coercive in the momentum variable, a case rarely considered in the classical literature about Hamilton-Jacobi equations.

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  • C. Esteve, H. Kouhkouh, D. Pighin, E. Zuazua. The Turnpike property and the long-time behavior of the Hamilton-Jacobi equation

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Author:Carlos Esteve, H Kouhkouh, Dario Pighin, Enrique Zuazua
Document Type:Article
Language:English
Date of Publication (online):2020/06/19
Date of first Publication:2020/10/27
Release Date:2020/10/27
Tag:Hamilton-Jacobi-Bellman equations; Optimal control problems; linear-quadratic; long-time behavior; the turnpike property
Subprojects:C03
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International