# American Institute of Mathematical Sciences

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

## On noncooperative $n$-player principal eigenvalue games

 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

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

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##### 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.
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