American Institute of Mathematical Sciences

Advanced
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 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.

show all references

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.

[1]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations and Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[2]

Jae Deok Kim, Ganguk Hwang. Cross-layer modeling and optimization of multi-channel cognitive radio networks under imperfect channel sensing. Journal of Industrial and Management Optimization, 2015, 11 (3) : 807-828. doi: 10.3934/jimo.2015.11.807
[3]

Zhanyou Ma, Wenbo Wang, Wuyi Yue, Yutaka Takahashi. Performance analysis and optimization research of multi-channel cognitive radio networks with a dynamic channel vacation scheme. Journal of Industrial and Management Optimization, 2022, 18 (1) : 95-110. doi: 10.3934/jimo.2020144
[4]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics and Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537
[5]

Aicha Balhag, Zaki Chbani, Hassan Riahi. Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evolution Equations and Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019
[6]

René Aïd, Roxana Dumitrescu, Peter Tankov. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics and Games, 2021, 8 (4) : 331-358. doi: 10.3934/jdg.2021012
[7]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[8]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[9]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177
[10]

Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027
[11]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure and Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254
[12]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics and Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[13]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[14]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[15]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217
[16]

M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137
[17]

Karthik Elamvazhuthi, Piyush Grover. Optimal transport over nonlinear systems via infinitesimal generators on graphs. Journal of Computational Dynamics, 2018, 5 (1&2) : 1-32. doi: 10.3934/jcd.2018001
[18]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315
[19]

Pablo Blanc. A lower bound for the principal eigenvalue of fully nonlinear elliptic operators. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3613-3623. doi: 10.3934/cpaa.2020158
[20]

Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106

 Impact Factor: 

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy