TY - JOUR

T1 - Strict dissipativity for discrete time discounted optimal control problems

AU - Grüne, Lars

AU - Müller, Matthias A.

AU - Kellett, Christopher M.

AU - Weller, Steven R.

N1 - Funding Information: 2020 Mathematics Subject Classification. Primary: 49J21, 93C55; Secondary: 93D20. Key words and phrases. Dissipativity, optimal control, discounting. The research was supported by the Australian Research Council under grants DP160102138 and DP180103026 and by the Deutsche Forschungsgemeinschaft under grant Gr1569/13-2. ∗ Corresponding author: Lars Grüne.

PY - 2021/12

Y1 - 2021/12

N2 - 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 wellknown results in the standard, undiscounted, setting whereby (strict) dissipa-tivity, 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 sup-ply 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 equi-libria. Several examples are provided throughout.

AB - 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 wellknown results in the standard, undiscounted, setting whereby (strict) dissipa-tivity, 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 sup-ply 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 equi-libria. Several examples are provided throughout.

KW - Discounting

KW - Dissipativity

KW - Optimal control

UR - http://www.scopus.com/inward/record.url?scp=85120830698&partnerID=8YFLogxK

U2 - 10.3934/MCRF.2020046

DO - 10.3934/MCRF.2020046

M3 - Article

AN - SCOPUS:85120830698

VL - 11

SP - 771

EP - 796

JO - Mathematical Control and Related Fields

JF - Mathematical Control and Related Fields

SN - 2156-8472

IS - 4

ER -