Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems

Veljko Ašković; Emmanuel Trélat; Hasnaa Zidani

doi:10.3934/mcrf.2024041

Article Contents

2024, Volume 14, Issue 4: 1501-1516. Doi: 10.3934/mcrf.2024041

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Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems

1.
MBDA, France
2.
Sorbonne Université, CNRS, Université Paris Cité, Inria, Laboratoire Jacques-Louis Lions (LJLL), Paris, F-75005, France
3.
INSA Rouen Normandie, Normandie Universtité, Laboratoire de Mathématiques (LMI), UR 3226, Rouen, F-76000, France

^*Corresponding author: Emmanuel Trélat
^*Corresponding author: Emmanuel Trélat

Received: December 2023

Revised: August 2024

Early access: August 30, 2024

Published online: August 30, 2024

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Abstract

Considering a general nonlinear dissipative finite dimensional optimal control problem in fixed time horizon $ T $, we establish a two-term asymptotic expansion of the value function as $ T\rightarrow +\infty $. The dominating term is $ T $ times the optimal value obtained from the optimal static problem within the classical turnpike theory. The second term, of order unity, is interpreted as the sum of two values associated with optimal stabilization problems related to the turnpike.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Turnpike illustration: (a) small time case (b) large time case

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Figure 2. Construction of a trajectory for the $ \mathcal{P}_{[0,T]}^{y_{0},y_{1}} $ problem

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