Strict dissipativity for discrete time discounted optimal control problems

Grüne Lars; Müller Matthias A.; Kellett Christopher M.; Weller Steven R.

doi:10.3934/mcrf.2020046

Article Contents

2021, Volume 11, Issue 4: 771-796. Doi: 10.3934/mcrf.2020046 open access

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Strict dissipativity for discrete time discounted optimal control problems

1.
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
2.
Institute of Automatic Control, Leibniz University Hannover, 30167 Hannover, Germany
3.
Research School of Electrical, Energy and Materials Engineering, Australian National University, Canberra, ACT 2600, Australia
4.
School of Electrical Engineering and Computing, University of Newcastle, Callaghan, NSW 2308, Australia

^* Corresponding author: Lars Grüne
^* Corresponding author: Lars Grüne

Received: August 31, 2019

Revised: April 30, 2020

Early access: November 2020

Published: December 2021

The research was supported by the Australian Research Council under grants DP160102138 and DP180103026 and by the Deutsche Forschungsgemeinschaft under grant Gr1569/13-2

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Abstract

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 available storage functions, required supply functions, 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.

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Mathematics Subject Classification: Primary: 49J21, 93C55; Secondary: 93D20.

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Figure 1. Illustration of the steady-states (blue solid line), level sets of $ \ell $ (black ellipses), and the additional constraint $ g_{ad} $ (red dashed) of the example in Section 8.4. The optimal steady-state $ (x^e,u^e)=(0,0) $ for the undiscounted case is marked with a circle

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