Strict dissipativity for discrete time discounted optimal control problems

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Grüne, Lars ; Müller, Matthias A. ; Kellett, Christopher M. ; Weller, Steven R.:
Strict dissipativity for discrete time discounted optimal control problems.
In: Mathematical Control and Related Fields. Vol. 11 (2021) Issue 4 . - pp. 771-796.
ISSN 2156-8472
DOI: https://doi.org/10.3934/mcrf.2020046

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The paradigm of discounting future costs is a common feature of economic applications of optimal control. In this paper, we provide several results for such discounted optimal control aimed at replicating the now well-known results in the standard, undiscounted, setting whereby (strict) dissipativity, turnpike properties, and near-optimality of closed-loop systems using model predictive control are essentially equivalent. To that end, we introduce a notion of discounted strict dissipativity and show that this implies various properties including the existence of an appropriate available storage function and robustness of optimal equilibria. Additionally, for discount factors sufficiently close to one we demonstrate that strict dissipativity implies discounted strict dissipativity and that optimally controlled systems, derived from a discounted cost function, yield practically asymptotically stable equilibria. Several examples are provided throughout.