Local turnpike analysis using local dissipativity for discrete time discounted optimal control

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Grüne, Lars ; Krügel, Lisa:
Local turnpike analysis using local dissipativity for discrete time discounted optimal control.
In: Applied Mathematics and Optimization. Bd. 84 (2021) Heft Suppl. 2 . - S. 1585-1606.
ISSN 1432-0606
DOI: https://doi.org/10.1007/s00245-021-09805-4

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Abstract

Recent results in the literature have provided connections between the so-called turnpike property, near optimality of closed-loop solutions, and strict dissipativity. Motivated by applications in economics, optimal control problems with discounted stage cost are of great interest. In contrast to non-discounted optimal control problems, it is more likely that several asymptotically stable optimal equilibria coexist. Due to the discounting and transition cost from a local to the global equilibrium, it may be more favourable staying in a local equilibrium than moving to the global - cheaper - equilibrium. In the literature, strict dissipativity was shown to provide criteria for global asymptotic stability of optimal equilibria and turnpike behavior. In this paper, we propose a local notion of discounted strict dissipativity and a local turnpike property, both depending on the discount factor. Using these concepts, we investigate the local behaviour of (near-)optimal trajectories and develop conditions on the discount factor to ensure convergence to a local asymptotically stable optimal equilibrium.