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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[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).

[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.

[21]

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

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[36]

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

[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.

[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.

[39]

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

[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.

[41]

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

[42]

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

[43]

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

[44]

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

[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.

[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.

[47]

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

[48]

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

[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.

[50]

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

[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.

[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.

[53]

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

[54]

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

[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.

[56]

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

[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.

[58]

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

[59]

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

[60]

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

[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.

[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.

[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.

[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.

[65]

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

[66]

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

[67]

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

[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.

[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.

[70]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[15]

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

[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.

[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.

[18]

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

[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).

[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.

[21]

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

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[35]

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

[36]

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

[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.

[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.

[39]

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

[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.

[41]

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

[42]

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

[43]

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

[44]

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

[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.

[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.

[47]

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

[48]

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

[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.

[50]

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

[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.

[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.

[53]

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

[54]

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

[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.

[56]

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

[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.

[58]

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

[59]

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

[60]

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

[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.

[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.

[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.

[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.

[65]

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

[66]

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

[67]

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

[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.

[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.

[70]

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

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