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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

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
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J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (1984).   Google Scholar

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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

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M. V. Day, Recent progress on the small parameter exit problem,, Stochastics, 20 (1987), 121.  doi: 10.1080/17442508708833440.  Google Scholar

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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

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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

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A. Friedman, Stochastic Differential Equations and Applications,, Vol. 2, (1976).   Google Scholar

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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

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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

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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

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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

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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