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arXiv · 2006.10430

The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems

Carlos Esteve·Hicham Kouhkouh·Dario Pighin·Enrique Zuazua
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Abstract

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 $T$ tends to infinity, the value function asymptotically behaves as $W(x) + c\, T + \lambda$, 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 showed that linear-quadratic optimal control problems with constrained control enjoy a turnpike property, also particularly when the steady optimum may saturate the control constraints.

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Carlos Esteve, Hicham Kouhkouh, Dario Pighin, Enrique Zuazua. 2020-06-18. The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems. https://arxiv.org/abs/2006.10430

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