2013, 3(1): 1-30. doi: 10.3934/naco.2013.3.1

Jamming in mobile networks: A game-theoretic approach

1. 

Department of Mechanical Engineering, Iowa State University, IA, 50011Ames, United States

2. 

Department of Aerospace Engineering, University of Illinois at Urbana Champaign, IL 61801, Urbana, United States

3. 

Department of Electrical and Computer Engineering and Coordinated Science Lab, University of Illinois at Urbana Champaign, IL 61801, Urbana, United States

Received  December 2011 Revised  November 2012 Published  January 2013

In this paper, we address the problem of jamming in a communication network within a team of mobile autonomous agents. In contradistinction with the contemporary research regarding jamming, we model the intrusion as a pursuit-evasion game between a mobile jammer and a team of agents.
     First, we consider a differential game-theoretic approach to compute optimal strategies for a team of UAVs trying to evade a jamming attack initiated by an aerial jammer in their vicinity. We formulate the problem as a zero-sum pursuit-evasion game, where the cost function is the termination time of the game. We use Isaacs' approach to obtain necessary conditions to arrive at the equations governing the saddle-point strategies of the players. We illustrate the results through simulations. Next, we analyze the problem of jamming from the perspective of maintaining connectivity in a network of mobile agents in the presence of an adversary. This is a variation of the standard connectivity maintenance problem in which the main issue is to deal with the limitations in communications and sensing model of each agent. In our work, the limitations in communication are due to the presence of a jammer in the vicinity of the mobile agents. We compute evasion strategies for the team of vehicles based on the connectivity of the resultant state-dependent graph. We present some simulations to validate the proposed control scheme. Finally, we address the problem of jamming for the scenario in which each agent computes its control strategy based on limited information available about its neighbors in the network. Under this decentralized information structure, we propose two approximation schemes for the agents and study the performance of the entire team for each scheme.
Citation: Sourabh Bhattacharya, Abhishek Gupta, Tamer Başar. Jamming in mobile networks: A game-theoretic approach. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 1-30. doi: 10.3934/naco.2013.3.1
References:
[1]

V. I. Arnold, "Geometric Method in the Theory of Ordinary Differential Equations,", Springer-Verlag, (1983).  doi: 10.1007/978-1-4684-0147-9.  Google Scholar

[2]

T. Başar, Two-criteria LQG decision problems with one-step delay observation sharing pattern,, Information and Control, 38 (1978), 21.  doi: 10.1016/S0019-9958(78)90018-9.  Google Scholar

[3]

T. Başar, On the saddle-point solution of a class of stochastic differential games,, Journal on Optimization Theory and Applications, 33 (1981), 539.  doi: 10.1007/BF00935757.  Google Scholar

[4]

T. Başar and S. Li, Distributed computation of Nash equilibria in linear-quadratic stochastic differential games,, SIAM Journal on Control and Optimization, 27 (1989), 563.  doi: 10.1137/0327030.  Google Scholar

[5]

T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory,", 2nd Ed., (1999).   Google Scholar

[6]

Y. Bar-Shalom and E. Tse, Dual effect, certainty equivalence, and separation in stochastic control,, IEEE Transactions on Automatic Control, 19 (1974), 494.  doi: 10.1109/TAC.1974.1100635.  Google Scholar

[7]

S. Bhattacharya, A. Gupta and T. Başar, Decentralized opportunistic navigation strategies for multi-agent systems in the presence of an adversary,, in, (2011), 11809.   Google Scholar

[8]

S. Bhattacharya and T. Başar, Differential game-theoretic approach for spatial jamming attack in a UAV communication network,, in, (2010).   Google Scholar

[9]

S Bhattacharya and T. Başar, Game-theoretic analysis of an aerial jamming attack on a UAVcommunication network ,, in, (2010), 818.   Google Scholar

[10]

S. Bhattacharya and T. Başar, Graph-theoretic approach to connectivity maintenance in mobile networks in the presence of a jammer,, in, (2010), 3560.   Google Scholar

[11]

S Bhattacharya and T. Başar, Optimal strategies to evade jamming in heterogeneous mobile networks,, in, (2010).   Google Scholar

[12]

S. Bhattacharya and T. Başar, Differential game-theoretic approach to a spatial jamming problem,, Annals of Dynamic Games, (2011).   Google Scholar

[13]

S. Bhattacharya and T. Başar, Spatial approaches to broadband jamming in heterogeneous mobile networks: a game-theoretic approach,, Autonomous Robots, 31 (2011), 367.  doi: 10.1007/s10514-011-9253-0.  Google Scholar

[14]

N. Biggs., "Algebraic Graph Theory,", Cambridge University Press, (1993).   Google Scholar

[15]

A. Blaquière, F. Gerard and G. Leitmann., "Quantitative and Qualitative Games,", Academic Press, (1969).   Google Scholar

[16]

J. V. Breakwell and P. Hagedorn, Further properties of non-zero sum differential games,, Journal of Optimization Theory and Applications, 3 (1969), 207.  doi: 10.1007/BF00926523.  Google Scholar

[17]

J. V. Breakwell and P. Hagedorn, Point capture of two evaders in succession,, Journal of Optimization Theory and Applications, 27 (1979), 89.  doi: 10.1007/BF00933327.  Google Scholar

[18]

H. Cao, E. Ertin, V. Kulathumani, M. Sridharan and A. Arora, Differential games in large-scale sensor-actuator networks,, in, (2006), 77.   Google Scholar

[19]

H. Choset, K.M. Lynch, S. Hutchinson, G. Kantor, W. Burgard, L. Kavraki and S. Thrun, "Principles of Robot Motion: Theory, Algorithms, and Implementations,", The MIT Press, (2005).   Google Scholar

[20]

R. Cogill and S. Lall, An approximation algorithm for the discrete team decision problem,, SIAM Journal on Control and Optimization, 45 (2007), 1359.  doi: 10.1137/050628374.  Google Scholar

[21]

M. C. DeGennaro and A. Jadbabaie, Decentralized control of connectivity for multiagent systems,, in, (2006), 3628.   Google Scholar

[22]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations,, IEEE Transactions on Automatic Control, 9 (2004), 1465.  doi: 10.1109/TAC.2004.834433.  Google Scholar

[23]

F. Bullo G. Notarstefano, K. Savla and A. Jadbabaie, Maintaining limited-range connectivity among second order agents,, in, (2006), 2124.   Google Scholar

[24]

S. Gorman, Y. J. Dreazen and A. Cole, Insurgents hack U.S. drones,, December 2009. Available from: , (2009).   Google Scholar

[25]

A. Gupta, S. Bhattacharya and T. Başar, Decentralized control of multi agent system with adversarial switching topology,, in, (2011).   Google Scholar

[26]

A. Jadbabaie, E. Stump and V. Kumar, Connectivity management in mobile robot teams,, in, (2008), 1525.   Google Scholar

[27]

A. Jadbabaie H. Tanner and G. Pappas, Flocking in fixed and switching networks,, IEEE Transactions on Automatic Control, 5 (2007), 863.   Google Scholar

[28]

P. Hagedorn and J. V. Breakwell, A differential game of approach with two pursuers and one evader,, Journal of Optimization Theory and Applications, 18 (1976), 15.  doi: 10.1007/BF00933791.  Google Scholar

[29]

R. Isaacs, "Differential Games,", Wiley, (1965).   Google Scholar

[30]

M. Ji and M. Egerstedt, Connectedness preserving distributed coordination control among dynamic graphs,, in, (2005), 93.   Google Scholar

[31]

M. Ji and M. Egerstedt, Distributed formation control while preserving connectedness,, in, (2006), 5962.   Google Scholar

[32]

G. Leitmann, An optimum pursuit problem,, Journal of the Franklin Institute, 263 (1957), 499.  doi: 10.1016/0016-0032(57)90227-2.  Google Scholar

[33]

G. Leitmann, "An Introduction to Optimal Control,", McGraw-Hill, (1966).   Google Scholar

[34]

G. Leitmann, A differential game of pursuit and evasion,, International Journal of Non Linear Mechanics, 4 (1969), 1.  doi: 10.1016/0020-7462(69)90008-0.  Google Scholar

[35]

G. Leitmann, "Cooperative and Non-Cooperative Many Player Differential Games,", Springer Verlag, (1974).   Google Scholar

[36]

G. Leitmann, Guaranteed avoidance feedback control,, IEEE Transactions on Automatic Control, 25 (1980), 850.  doi: 10.1109/TAC.1980.1102408.  Google Scholar

[37]

G. Leitmann and S. Gutman, Optimal strategies in the neighborhood of a collision course,, AIAA Journal, 14 (1976), 1210.  doi: 10.2514/3.7213.  Google Scholar

[38]

A. Y. Levchenkov and A. G. Pashkov, Differential game of optimal approach of two inertial pursuers to a noninertial evader,, Journal of Optimization Theory and Applications, 65 (1990), 501.  doi: 10.1007/BF00939563.  Google Scholar

[39]

J. Lewin, "Differential Games: Theory and Methods for Solving Game Problems with Singular Surfaces,", Springer-Verlag, (1994).   Google Scholar

[40]

S. Li and T. Başar, Distributed algorithms for the computation of noncooperative equilibria,, Automatica, 23 (1987), 523.  doi: 10.1016/0005-1098(87)90081-1.  Google Scholar

[41]

D. Liberzon, "Switching in Systems and Control,", Birkhauser, (2003).  doi: 10.1007/978-1-4612-0017-8.  Google Scholar

[42]

D. McCullagh, Predator drones hacked in Iraq operations,, December 2009, (2009), 8301.   Google Scholar

[43]

A. A. Melikyan, "Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games,", Applications of Mathematics, (2000).   Google Scholar

[44]

M. Mesbahi and M. Egerstedt, "Graph Theoretic Methods in Multiagent Networks,", Princeton University Press, (2010).   Google Scholar

[45]

Mehran Mesbahi, On state-dependent dynamic graphs and their controllability properties,, IEEE Transactions on Automatic Control, 50 (2005), 387.  doi: 10.1109/TAC.2005.843858.  Google Scholar

[46]

I. M. Mitchell and C. J. Tomlin, Overapproximating reachable sets by hamilton-jacobi projections,, Journal of Scientific Computing, 19 (2003), 323.  doi: 10.1023/A:1025364227563.  Google Scholar

[47]

V. Kumar N. Michael, M. M. Zavlanos and G. J. Pappas, Maintaining connectivity in mobile robot networks,, in, (2008).   Google Scholar

[48]

G. Noubir and G. Lin, Low power denial of service attacks in data wireless lans and countermeasures,, in, (2003).   Google Scholar

[49]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time delay,, IEEE Transactions on Automatic Control, 49 (2004), 1520.  doi: 10.1109/TAC.2004.834113.  Google Scholar

[50]

P. Papadimitratos and Z. J. Haas, Secure routing for mobile ad hoc networks,, in, (2002), 27.   Google Scholar

[51]

C. H. Papadimitriou and J. N. Tsitsiklis, Intractable problems in control theory,, SIAM journal on control and optimization, 24 (1986), 639.  doi: 10.1137/0324038.  Google Scholar

[52]

A. G. Pashkov and S. D. Terekhov, A differential game of approach with two pursuers and one evader,, Journal of Optimization Theory and Applications, 55 (1987), 303.  doi: 10.1007/BF00939087.  Google Scholar

[53]

R. A. Poisel, "Modern Communication Jamming Principles and Techniques,", Artech, (2004).   Google Scholar

[54]

John J. Proakis and Masoud Salehi, "Digital Communications,", McGraw-Hill, (2007).   Google Scholar

[55]

I. Rhodes and D. Luenberger, Differential games with imperfect state information,, IEEE Transactions on Automatic Control, 14 (1969), 29.  doi: 10.1109/TAC.1969.1099086.  Google Scholar

[56]

Sriram Shankaran, Dušsan Stipanović and Claire Tomlin, Collision avoidance strategies for a three player game,, in, (2008).   Google Scholar

[57]

J. Shinar and T. Vladimir, What happens when certainty equivalence is not valid? Is there an optimal estimator for terminal guidance?,, Annual Reviews in Control, 27 (2003), 119.  doi: 10.1016/j.arcontrol.2003.10.001.  Google Scholar

[58]

D. P. Spanos and R. M. Murray, Robust connectivity of networked vehicles,, in, (2004), 2893.   Google Scholar

[59]

M. Spivak, "Calculus on Manifolds: A modern approach to Classical Theorems of Advanced Calculus,", Perseus Books Publishing, (1965).   Google Scholar

[60]

D. M. Stipanović, S. Shankaran and C. Tomlin, Strategies for agents in multi-player pursuit-evasion games,, in, (2008).   Google Scholar

[61]

D. M. Stipanović, I. Hwang and C. J. Tomlin, Computation of an over-approximation of the backward reachable set using subsystem level set functions,, Dynamics of Continuous, 11 (2004), 399.   Google Scholar

[62]

D. M. Stipanović, A. A. Melikyan and N. V. Hovakimyan, Some sufficient conditions for multi-player pursuit evasion games with continuous and discrete observations,, in, (2009), 133.   Google Scholar

[63]

J. Tobias and N. Seddon, Signal jamming mediates sexual conflict in a duetting bird,, Current Biology, 19 (2009), 577.  doi: 10.1016/j.cub.2009.02.036.  Google Scholar

[64]

J. Tsitsiklis and M. Athans, On the complexity of decentralized decision making and detection problems,, IEEE Transactions on Automatic Control, 30 (1985), 440.  doi: 10.1109/TAC.1985.1103988.  Google Scholar

[65]

E. M. Vaisbord and V. I. Zhukovskiy, "Introduction to Multi-player Differential Games and their Applications,", Gordon and Breach, (1988).   Google Scholar

[66]

M. M. Zavlanos and G. J. Pappas, Controlling connectivity of dynamic networks,, in, (2005), 6388.  doi: 10.1109/CDC.2005.1583186.  Google Scholar

[67]

M. M. Zavlanos and G. J. Pappas, Distributed connectivity control of mobile networks,, in, (2007), 3591.  doi: 10.1109/CDC.2007.4434525.  Google Scholar

[68]

M. M. Zavlanos and G. J. Pappas, Potential fields for maintaining connectivity of mobile networks,, IEEE Transactions on Robotics, 23 (2007), 812.  doi: 10.1109/TRO.2007.900642.  Google Scholar

[69]

M. M. Zavlanos and G. J. Pappas, Distributed connectivity control of mobile networks,, IEEE Transactions on Robotic, 24 (2008), 1416.  doi: 10.1109/TRO.2008.2006233.  Google Scholar

[70]

V. I. Zhukovskiy and M. E. Salukvadze, "The Vector Valued Maxmin,", Academic Press, (1994).   Google Scholar

show all references

References:
[1]

V. I. Arnold, "Geometric Method in the Theory of Ordinary Differential Equations,", Springer-Verlag, (1983).  doi: 10.1007/978-1-4684-0147-9.  Google Scholar

[2]

T. Başar, Two-criteria LQG decision problems with one-step delay observation sharing pattern,, Information and Control, 38 (1978), 21.  doi: 10.1016/S0019-9958(78)90018-9.  Google Scholar

[3]

T. Başar, On the saddle-point solution of a class of stochastic differential games,, Journal on Optimization Theory and Applications, 33 (1981), 539.  doi: 10.1007/BF00935757.  Google Scholar

[4]

T. Başar and S. Li, Distributed computation of Nash equilibria in linear-quadratic stochastic differential games,, SIAM Journal on Control and Optimization, 27 (1989), 563.  doi: 10.1137/0327030.  Google Scholar

[5]

T. Başar and G. J. Olsder, "Dynamic Noncooperative Game Theory,", 2nd Ed., (1999).   Google Scholar

[6]

Y. Bar-Shalom and E. Tse, Dual effect, certainty equivalence, and separation in stochastic control,, IEEE Transactions on Automatic Control, 19 (1974), 494.  doi: 10.1109/TAC.1974.1100635.  Google Scholar

[7]

S. Bhattacharya, A. Gupta and T. Başar, Decentralized opportunistic navigation strategies for multi-agent systems in the presence of an adversary,, in, (2011), 11809.   Google Scholar

[8]

S. Bhattacharya and T. Başar, Differential game-theoretic approach for spatial jamming attack in a UAV communication network,, in, (2010).   Google Scholar

[9]

S Bhattacharya and T. Başar, Game-theoretic analysis of an aerial jamming attack on a UAVcommunication network ,, in, (2010), 818.   Google Scholar

[10]

S. Bhattacharya and T. Başar, Graph-theoretic approach to connectivity maintenance in mobile networks in the presence of a jammer,, in, (2010), 3560.   Google Scholar

[11]

S Bhattacharya and T. Başar, Optimal strategies to evade jamming in heterogeneous mobile networks,, in, (2010).   Google Scholar

[12]

S. Bhattacharya and T. Başar, Differential game-theoretic approach to a spatial jamming problem,, Annals of Dynamic Games, (2011).   Google Scholar

[13]

S. Bhattacharya and T. Başar, Spatial approaches to broadband jamming in heterogeneous mobile networks: a game-theoretic approach,, Autonomous Robots, 31 (2011), 367.  doi: 10.1007/s10514-011-9253-0.  Google Scholar

[14]

N. Biggs., "Algebraic Graph Theory,", Cambridge University Press, (1993).   Google Scholar

[15]

A. Blaquière, F. Gerard and G. Leitmann., "Quantitative and Qualitative Games,", Academic Press, (1969).   Google Scholar

[16]

J. V. Breakwell and P. Hagedorn, Further properties of non-zero sum differential games,, Journal of Optimization Theory and Applications, 3 (1969), 207.  doi: 10.1007/BF00926523.  Google Scholar

[17]

J. V. Breakwell and P. Hagedorn, Point capture of two evaders in succession,, Journal of Optimization Theory and Applications, 27 (1979), 89.  doi: 10.1007/BF00933327.  Google Scholar

[18]

H. Cao, E. Ertin, V. Kulathumani, M. Sridharan and A. Arora, Differential games in large-scale sensor-actuator networks,, in, (2006), 77.   Google Scholar

[19]

H. Choset, K.M. Lynch, S. Hutchinson, G. Kantor, W. Burgard, L. Kavraki and S. Thrun, "Principles of Robot Motion: Theory, Algorithms, and Implementations,", The MIT Press, (2005).   Google Scholar

[20]

R. Cogill and S. Lall, An approximation algorithm for the discrete team decision problem,, SIAM Journal on Control and Optimization, 45 (2007), 1359.  doi: 10.1137/050628374.  Google Scholar

[21]

M. C. DeGennaro and A. Jadbabaie, Decentralized control of connectivity for multiagent systems,, in, (2006), 3628.   Google Scholar

[22]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations,, IEEE Transactions on Automatic Control, 9 (2004), 1465.  doi: 10.1109/TAC.2004.834433.  Google Scholar

[23]

F. Bullo G. Notarstefano, K. Savla and A. Jadbabaie, Maintaining limited-range connectivity among second order agents,, in, (2006), 2124.   Google Scholar

[24]

S. Gorman, Y. J. Dreazen and A. Cole, Insurgents hack U.S. drones,, December 2009. Available from: , (2009).   Google Scholar

[25]

A. Gupta, S. Bhattacharya and T. Başar, Decentralized control of multi agent system with adversarial switching topology,, in, (2011).   Google Scholar

[26]

A. Jadbabaie, E. Stump and V. Kumar, Connectivity management in mobile robot teams,, in, (2008), 1525.   Google Scholar

[27]

A. Jadbabaie H. Tanner and G. Pappas, Flocking in fixed and switching networks,, IEEE Transactions on Automatic Control, 5 (2007), 863.   Google Scholar

[28]

P. Hagedorn and J. V. Breakwell, A differential game of approach with two pursuers and one evader,, Journal of Optimization Theory and Applications, 18 (1976), 15.  doi: 10.1007/BF00933791.  Google Scholar

[29]

R. Isaacs, "Differential Games,", Wiley, (1965).   Google Scholar

[30]

M. Ji and M. Egerstedt, Connectedness preserving distributed coordination control among dynamic graphs,, in, (2005), 93.   Google Scholar

[31]

M. Ji and M. Egerstedt, Distributed formation control while preserving connectedness,, in, (2006), 5962.   Google Scholar

[32]

G. Leitmann, An optimum pursuit problem,, Journal of the Franklin Institute, 263 (1957), 499.  doi: 10.1016/0016-0032(57)90227-2.  Google Scholar

[33]

G. Leitmann, "An Introduction to Optimal Control,", McGraw-Hill, (1966).   Google Scholar

[34]

G. Leitmann, A differential game of pursuit and evasion,, International Journal of Non Linear Mechanics, 4 (1969), 1.  doi: 10.1016/0020-7462(69)90008-0.  Google Scholar

[35]

G. Leitmann, "Cooperative and Non-Cooperative Many Player Differential Games,", Springer Verlag, (1974).   Google Scholar

[36]

G. Leitmann, Guaranteed avoidance feedback control,, IEEE Transactions on Automatic Control, 25 (1980), 850.  doi: 10.1109/TAC.1980.1102408.  Google Scholar

[37]

G. Leitmann and S. Gutman, Optimal strategies in the neighborhood of a collision course,, AIAA Journal, 14 (1976), 1210.  doi: 10.2514/3.7213.  Google Scholar

[38]

A. Y. Levchenkov and A. G. Pashkov, Differential game of optimal approach of two inertial pursuers to a noninertial evader,, Journal of Optimization Theory and Applications, 65 (1990), 501.  doi: 10.1007/BF00939563.  Google Scholar

[39]

J. Lewin, "Differential Games: Theory and Methods for Solving Game Problems with Singular Surfaces,", Springer-Verlag, (1994).   Google Scholar

[40]

S. Li and T. Başar, Distributed algorithms for the computation of noncooperative equilibria,, Automatica, 23 (1987), 523.  doi: 10.1016/0005-1098(87)90081-1.  Google Scholar

[41]

D. Liberzon, "Switching in Systems and Control,", Birkhauser, (2003).  doi: 10.1007/978-1-4612-0017-8.  Google Scholar

[42]

D. McCullagh, Predator drones hacked in Iraq operations,, December 2009, (2009), 8301.   Google Scholar

[43]

A. A. Melikyan, "Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games,", Applications of Mathematics, (2000).   Google Scholar

[44]

M. Mesbahi and M. Egerstedt, "Graph Theoretic Methods in Multiagent Networks,", Princeton University Press, (2010).   Google Scholar

[45]

Mehran Mesbahi, On state-dependent dynamic graphs and their controllability properties,, IEEE Transactions on Automatic Control, 50 (2005), 387.  doi: 10.1109/TAC.2005.843858.  Google Scholar

[46]

I. M. Mitchell and C. J. Tomlin, Overapproximating reachable sets by hamilton-jacobi projections,, Journal of Scientific Computing, 19 (2003), 323.  doi: 10.1023/A:1025364227563.  Google Scholar

[47]

V. Kumar N. Michael, M. M. Zavlanos and G. J. Pappas, Maintaining connectivity in mobile robot networks,, in, (2008).   Google Scholar

[48]

G. Noubir and G. Lin, Low power denial of service attacks in data wireless lans and countermeasures,, in, (2003).   Google Scholar

[49]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time delay,, IEEE Transactions on Automatic Control, 49 (2004), 1520.  doi: 10.1109/TAC.2004.834113.  Google Scholar

[50]

P. Papadimitratos and Z. J. Haas, Secure routing for mobile ad hoc networks,, in, (2002), 27.   Google Scholar

[51]

C. H. Papadimitriou and J. N. Tsitsiklis, Intractable problems in control theory,, SIAM journal on control and optimization, 24 (1986), 639.  doi: 10.1137/0324038.  Google Scholar

[52]

A. G. Pashkov and S. D. Terekhov, A differential game of approach with two pursuers and one evader,, Journal of Optimization Theory and Applications, 55 (1987), 303.  doi: 10.1007/BF00939087.  Google Scholar

[53]

R. A. Poisel, "Modern Communication Jamming Principles and Techniques,", Artech, (2004).   Google Scholar

[54]

John J. Proakis and Masoud Salehi, "Digital Communications,", McGraw-Hill, (2007).   Google Scholar

[55]

I. Rhodes and D. Luenberger, Differential games with imperfect state information,, IEEE Transactions on Automatic Control, 14 (1969), 29.  doi: 10.1109/TAC.1969.1099086.  Google Scholar

[56]

Sriram Shankaran, Dušsan Stipanović and Claire Tomlin, Collision avoidance strategies for a three player game,, in, (2008).   Google Scholar

[57]

J. Shinar and T. Vladimir, What happens when certainty equivalence is not valid? Is there an optimal estimator for terminal guidance?,, Annual Reviews in Control, 27 (2003), 119.  doi: 10.1016/j.arcontrol.2003.10.001.  Google Scholar

[58]

D. P. Spanos and R. M. Murray, Robust connectivity of networked vehicles,, in, (2004), 2893.   Google Scholar

[59]

M. Spivak, "Calculus on Manifolds: A modern approach to Classical Theorems of Advanced Calculus,", Perseus Books Publishing, (1965).   Google Scholar

[60]

D. M. Stipanović, S. Shankaran and C. Tomlin, Strategies for agents in multi-player pursuit-evasion games,, in, (2008).   Google Scholar

[61]

D. M. Stipanović, I. Hwang and C. J. Tomlin, Computation of an over-approximation of the backward reachable set using subsystem level set functions,, Dynamics of Continuous, 11 (2004), 399.   Google Scholar

[62]

D. M. Stipanović, A. A. Melikyan and N. V. Hovakimyan, Some sufficient conditions for multi-player pursuit evasion games with continuous and discrete observations,, in, (2009), 133.   Google Scholar

[63]

J. Tobias and N. Seddon, Signal jamming mediates sexual conflict in a duetting bird,, Current Biology, 19 (2009), 577.  doi: 10.1016/j.cub.2009.02.036.  Google Scholar

[64]

J. Tsitsiklis and M. Athans, On the complexity of decentralized decision making and detection problems,, IEEE Transactions on Automatic Control, 30 (1985), 440.  doi: 10.1109/TAC.1985.1103988.  Google Scholar

[65]

E. M. Vaisbord and V. I. Zhukovskiy, "Introduction to Multi-player Differential Games and their Applications,", Gordon and Breach, (1988).   Google Scholar

[66]

M. M. Zavlanos and G. J. Pappas, Controlling connectivity of dynamic networks,, in, (2005), 6388.  doi: 10.1109/CDC.2005.1583186.  Google Scholar

[67]

M. M. Zavlanos and G. J. Pappas, Distributed connectivity control of mobile networks,, in, (2007), 3591.  doi: 10.1109/CDC.2007.4434525.  Google Scholar

[68]

M. M. Zavlanos and G. J. Pappas, Potential fields for maintaining connectivity of mobile networks,, IEEE Transactions on Robotics, 23 (2007), 812.  doi: 10.1109/TRO.2007.900642.  Google Scholar

[69]

M. M. Zavlanos and G. J. Pappas, Distributed connectivity control of mobile networks,, IEEE Transactions on Robotic, 24 (2008), 1416.  doi: 10.1109/TRO.2008.2006233.  Google Scholar

[70]

V. I. Zhukovskiy and M. E. Salukvadze, "The Vector Valued Maxmin,", Academic Press, (1994).   Google Scholar

[1]

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