On noncooperative n-player principal eigenvalue games

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January 2015, 2(1): 51-63. doi: 10.3934/jdg.2015.2.51

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, United States, United States

Received November 2014 Revised February 2015 Published June 2015

Abstract
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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: Asymptotic exit rate, principal eigenvalue, Nash equilibrium, multi-channel dynamical systems, noncooperative game., infinitesimal generator.

Mathematics Subject Classification: Primary: 34D10, 34D20, 47H10, 91A10, 93B6.

Citation: Getachew K. Befekadu, Panos J. Antsaklis. On noncooperative $n$-player principal eigenvalue games. Journal of Dynamics & Games, 2015, 2 (1) : 51-63. doi: 10.3934/jdg.2015.2.51

References:

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show all references

References:

[1]	J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (1984). Google Scholar
[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. doi: 10.1007/s00498-012-0105-z. Google Scholar
[3]	M. V. Day, On the exponential exit law in the small parameter exit problem,, Stochastics, 8 (1983), 297. doi: 10.1080/17442508308833244. Google Scholar
[4]	M. V. Day, Recent progress on the small parameter exit problem,, Stochastics, 20 (1987), 121. doi: 10.1080/17442508708833440. Google Scholar
[5]	M. V. Day and T. A. Darden, Some regularity results on the Ventcel-Freidlin quasipotential function,, Appl. Math. Optim., 13 (1985), 259. doi: 10.1007/BF01442211. Google Scholar
[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. doi: 10.1512/iumj.1978.27.27012. Google Scholar
[7]	A. Friedman, Stochastic Differential Equations and Applications,, Vol. 2, (1976). Google Scholar
[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. Google Scholar
[9]	Y. Kifer, On the principal eigenvalue in a singular perturbation problem with hyperbolic limit points and circles,, J. Diff. Equ., 37 (1980), 108. doi: 10.1016/0022-0396(80)90092-3. Google Scholar
[10]	Y. Kifer, The inverse problem for small random perturbations of dynamical systems,, Israel J. Math., 40 (1981), 165. doi: 10.1007/BF02761907. Google Scholar
[11]	S. J. Sheu, Some estimates of the transition density of a non-degenerate diffusion Markov process,, Ann. Probab., 19 (1991), 538. doi: 10.1214/aop/1176990440. Google Scholar
[12]	A. D. Ventcel and M. I. Freidlin, On small random perturbations of dynamical systems,, Russian Math. Surveys, 25 (1970), 3. Google Scholar
[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. doi: 10.1137/1120064. Google Scholar
[14]	A. D. Ventcel, Rough limit theorems on large deviations for Markov stochastic processes. I,, Theo. Prob. Appl., 21 (1977), 817. doi: 10.1137/1121030. Google Scholar

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