On noncooperative $n$-player principal eigenvalue games

Getachew K.  Befekadu; Panos J.  Antsaklis

doi:10.3934/jdg.2015.2.51

Article Contents

2015, Volume 2, Issue 1: 51-63. Doi: 10.3934/jdg.2015.2.51

This issue Previous Article Hamiltonian evolutionary games Next Article Discrete time dynamic oligopolies with adjustment constraints

On noncooperative $n$-player principal eigenvalue games

Getachew K. Befekadu^1, and
Panos J. Antsaklis^1,

1.
Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556

Received: November 2014

Revised: February 2015

Published: January 2015

Abstract Related Papers Cited by

Abstract

We consider a noncooperative $n$-player principal eigenvalue game which is associated with an infinitesimal generator of a stochastically perturbed multi-channel dynamical system -- where, in the course of such a game, each player attempts to minimize the asymptotic rate with which the controlled state trajectory of the system exits from a given bounded open domain. In particular, we show the existence of a Nash-equilibrium point (i.e., an $n$-tuple of equilibrium linear feedback operators) in a game-theoretic setting that is connected to a maximum closed invariant set of the corresponding deterministic multi-channel dynamical system, when the latter is composed with this $n$-tuple of equilibrium linear feedback operators.

Keywords:

Mathematics Subject Classification: Primary: 34D10, 34D20, 47H10, 91A10, 93B60.

Citation:

Related Papers

Cited by

References

[1]	J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.
[2]	G. K. Befekadu, V. Gupta and P. J. Antsaklis, Characterization of feedback Nash equilibria for multi-channel systems via a set of non-fragile stabilizing state-feedback solutions and dissipativity inequalities, J. Math. Contr. Sign. Syst., 25 (2013), 311-326.doi: 10.1007/s00498-012-0105-z.
[3]	M. V. Day, On the exponential exit law in the small parameter exit problem, Stochastics, 8 (1983), 297-323.doi: 10.1080/17442508308833244.
[4]	M. V. Day, Recent progress on the small parameter exit problem, Stochastics, 20 (1987), 121-150.doi: 10.1080/17442508708833440.
[5]	M. V. Day and T. A. Darden, Some regularity results on the Ventcel-Freidlin quasipotential function, Appl. Math. Optim., 13 (1985), 259-282.doi: 10.1007/BF01442211.
[6]	A. Devinatz and A. Friedman, Asymptotic behavior of the principal eigenfunction for a singularly perturbed Dirichlet problem, J. Indiana Univ. Math., 27 (1978), 143-157.doi: 10.1512/iumj.1978.27.27012.
[7]	A. Friedman, Stochastic Differential Equations and Applications, Vol. 2, Academic Press, 1976.
[8]	I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.
[9]	Y. Kifer, On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles, J. Diff. Equ., 37 (1980), 108-139.doi: 10.1016/0022-0396(80)90092-3.
[10]	Y. Kifer, The inverse problem for small random perturbations of dynamical systems, Israel J. Math., 40 (1981), 165-174.doi: 10.1007/BF02761907.
[11]	S. J. Sheu, Some estimates of the transition density of a non-degenerate diffusion Markov process, Ann. Probab., 19 (1991), 538-561.doi: 10.1214/aop/1176990440.
[12]	A. D. Ventcel and M. I. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys, 25 (1970), 3-55.
[13]	A. D. Ventcel, On the asymptotic behavior of the largest eigenvalue of a second-order elliptic differential operator with smaller parameter in the higher derivatives, Theo. Prob. Appl., 20 (1976), 599-602.doi: 10.1137/1120064.
[14]	A. D. Ventcel, Rough limit theorems on large deviations for Markov stochastic processes. I, Theo. Prob. Appl., 21 (1977), 817-821.doi: 10.1137/1121030.