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Robust Dynamic Programming in N Players Uncertain Differential Games
Volume 31, Issue 4 (2020), pp. 769–791
Pub. online: 23 November 2020      Type: Research Article      Open accessOpen Access
Received
1 July 2019
Accepted
1 November 2020
Published
23 November 2020

Abstract

In this paper we consider a non-cooperative N players differential game affected by deterministic uncertainties. Sufficient conditions for the existence of a robust feedback Nash equilibrium are presented in a set of min-max forms of Hamilton–Jacobi–Bellman equations. Such conditions are then used to find the robust Nash controls for a linear affine quadratic game affected by a square integrable uncertainty, which is seen as a malicious fictitious player trying to maximize the cost function of each player. The approach allows us to find robust strategies in the solution of a group of coupled Riccati differential equation. The finite, as well as infinite, time horizon cases are solved for this last game. As an illustration of the approach, the problem of the coordination of a two-echelon supply chain with seasonal uncertain fluctuations in demand is developed.

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Biographies

Jiménez-Lizárraga Manuel
manuel.jimenezlzr@uanl.edu.mx

M. Jimenez-Lizarraga received his BSc in electrical engineering from Culiacan Institute of Technology, and the MSc and PhD degrees in automatic control from CINVESTAV-IPN, in Mexico, in 1996, 2000, and 2006, respectively. He was a postdoctoral fellow at the ECE Department of the Ohio State University in 2009 and a visiting scholar at Heuristique el Diagnostic des Systemes Complexes at Universite de Technologie de Compiegne in 2015. He is a member of the Mexican Academy of Sciences and is currently with the Faculty of Physical and Mathematical Sciences of the UANL. His research interests include differential games, robust, optimal and sliding mode control and applications such as UAVs guidance and control.

Rodríguez-Sánchez Sara V.
sara.rodriguezsn@uanl.edu.mx

S.V. Rodriguez-Sanchez is an associate professor at the UANL. She received her BSc degree in industrial engineering from the Universidad Autonoma del Estado de Hidalgo, Mexico, in 2002 and her PhD degree in engineering from the Universidad de Lleida, Spain, in 2010. Her current research interests include the application of operation research and analytics in agroindustry and forestry.

de la Cruz Naín
nca200881@gmail.com

N. de la Cruz received his master in industrial physics engineering in 2017 from the Faculty of Physical and Mathematical Sciences of the Autonomous University of Nuevo Leon. His reseach interest include dynamic games and applied mathematics.

Villarreal César Emilio
cesar.villarrealrd@uanl.edu.mx

C.E. Villarreal received his doctoral degree in mathematics from the Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional (CINVESTAV-IPN), Mexico, in 1998. He is a professor at the Universidad Autonoma de Nuevo Leon (UANL), Mexico. His research interests include stochastic processes, probability and ergodic theory.


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Open access article under the CC BY license.

Keywords
differential games LQ games dynamic supply chain

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