Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence

Valery Y. Glizer; Oleg Kelis

doi:10.3934/naco.2017001

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2017, Volume 7, Issue 1: 1-20. Doi: 10.3934/naco.2017001

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Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence

Valery Y. Glizer^1, and
Oleg Kelis^2,3,

1.
Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel
2.
Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.Box 78, Karmiel 2161002, Israel
3.
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel

The first author is the corresponding author.

Received: September 2016

Published: May 2017

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Abstract

We consider an infinite horizon zero-sum linear-quadratic differential game in the case where the cost functional does not contain a control cost of the minimizing player (the minimizer). This feature means that the game under consideration is singular. For this game, novel definitions of the saddle-point equilibrium and game value are proposed. To obtain these saddle-point equilibrium and game value, we associate the singular game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. Using the solvability conditions, the solution of the cheap control game is reduced to solution of a Riccati matrix algebraic equation with an indefinite quadratic term. This equation is perturbed by a small parameter. Subject to a proper assumption, an asymptotic expansion of a stabilizing solution to this equation is constructed and justified. Using this asymptotic expansion, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.

Keywords:

Singular differential game,
infinite horizon zero-sum differential game,
saddle-point equilibrium,
game value,
regularization,
cheap control game,
perturbation,
Riccati matrix algebraic equation,
asymptotic solution.

Mathematics Subject Classification: Primary: 49N70, 91A05, 91A10, 91A23; Secondary: 15A24, 93C73.

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