Robust open-loop Nash equilibria in the noncooperative LQ game revisited

Jacob Engwerda

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.
Original languageEnglish
Pages (from-to)795-813
JournalOptimal Control Applications & Methods
Volume38
Issue number5
Early online date17 Nov 2016
DOIs
Publication statusPublished - Sep 2017

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Differential Games
Nash Equilibrium
Differential equations
Stabilization
Game
Planning
Quadratic Differentials
Riccati Differential Equation
Necessary
Information Structure
State Constraints
Control Policy
Solvability
Horizon
Uniqueness
Demonstrate

Keywords

  • robust control
  • noncooperative differential games
  • linear optimal control
  • Riccati equations

Cite this

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abstract = "This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.",
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Robust open-loop Nash equilibria in the noncooperative LQ game revisited. / Engwerda, Jacob.

In: Optimal Control Applications & Methods, Vol. 38, No. 5, 09.2017, p. 795-813.

Research output: Contribution to journalArticleScientificpeer-review

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AB - This paper reconsiders existence of worst-case Nash equilibria in noncooperative multi-player differential games, this, within an open-loop information structure. We show that these equilibria can be obtained by determining the open-loop Nash equilibria of an associated differential game with an additional initial state constraint. For the special case of linear-quadratic differential games, we derive both necessary and sufficient conditions for solvability of the finite planning horizon problem. In particular, we demonstrate that, unlike in the standard linear-quadratic differential game setting, uniqueness of equilibria may fail to hold. A both necessary and sufficient condition under which there is a unique equilibrium is provided. A sufficient existence condition for a unique equilibrium is derived in terms of a Riccati differential equation. Consequences for control policies are demonstrated in a simple debt stabilization game.

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