American Institute of Mathematical Sciences

Advanced
2011, 1(3): 487-493. doi: 10.3934/naco.2011.1.487

A note on monotone approximations of minimum and maximum functions and multi-objective problems

Dušan M. Stipanović 1, , Claire J. Tomlin 2, and  George Leitmann 3,

1. 

Coordinated Science Laboratory, Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, United States
2. 

Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, California, United States
3. 

College of Engineering, University of California at Berkeley, Berkeley, California, United States

Received  April 2011 Revised  July 2011 Published  September 2011

In paper [12] the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are formulated using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example of a capture of two evaders by two pursuers is provided.
    This note presents a preview of the treatment in [12].
Keywords: minimum function, multiple objectives, Dynamic systems, approximations of functions., maximum function.
Mathematics Subject Classification: Primary: 26D07, 34A40; Secondary: 34D2.
Citation: Dušan M. Stipanović, Claire J. Tomlin, George Leitmann. A note on monotone approximations of minimum and maximum functions and multi-objective problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487
References:
[1]

A. Bacciotti and L. Rosier, "Liapunov Functions and Stability in Control Theory," 2nd edition, Springer-Verlag, Berlin, Germany, 2005.  Google Scholar

[2]

R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques," Birkhäuser, Boston, MA, 1996. Google Scholar

[3]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover Publications, New York, NY, 1975.  Google Scholar

[4]

G. Leitmann and J. Skowronski, Avoidance control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298.  Google Scholar

[5]

G. Leitmann, Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, 32 (1980), 569-576. doi: 10.1007/BF00934040.  Google Scholar

[6]

G. Leitmann and J. Skowronski, A note on avoidance control, Optimal Control Applications & Methods, 4 (1983), 335-342. doi: 10.1002/oca.4660040406.  Google Scholar

[7]

K. M. Miettinen, "Nonlinear Multiobjective Optimization," Kluwer Academic Publishers, Norwell, MA, 1999. doi: 10.1007/978-1-4615-5563-6.  Google Scholar

[8]

D. M. Stipanović, Sriram and C. J. Tomlin, Strategies for agents in multi-player pursuit-evasion games, in "Proceedings of the Eleventh International Symposium on Dynamic Games and Applications," Tucson, Arizona, 2004. Google Scholar

[9]

D. M. Stipanović, P. F. Hokayem, M. W. Spong and D. D. Šiljak, Cooperative avoidance control for multi-agent systems, Journal of Dynamic Systems, Measurement and Control, 129 (2007), 699-707. Google Scholar

[10]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations, Annals of Dynamic Games, 10 (2009), 133-145. Google Scholar

[11]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Guaranteed strategies for nonlinear multi-player pursuit-evasion games, International Game Theory Review, 12 (2010), 1-17. Google Scholar

[12]

D. M. Stipanović, C. J. Tomlin and G. Leitmann, Monotone approximations of minimum and maximum functions and multi-objective problems,, Submitted to Applied Mathematics & Optimization., ().   Google Scholar

show all references

References:
[1]

A. Bacciotti and L. Rosier, "Liapunov Functions and Stability in Control Theory," 2nd edition, Springer-Verlag, Berlin, Germany, 2005.  Google Scholar

[2]

R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques," Birkhäuser, Boston, MA, 1996. Google Scholar

[3]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover Publications, New York, NY, 1975.  Google Scholar

[4]

G. Leitmann and J. Skowronski, Avoidance control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298.  Google Scholar

[5]

G. Leitmann, Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, 32 (1980), 569-576. doi: 10.1007/BF00934040.  Google Scholar

[6]

G. Leitmann and J. Skowronski, A note on avoidance control, Optimal Control Applications & Methods, 4 (1983), 335-342. doi: 10.1002/oca.4660040406.  Google Scholar

[7]

K. M. Miettinen, "Nonlinear Multiobjective Optimization," Kluwer Academic Publishers, Norwell, MA, 1999. doi: 10.1007/978-1-4615-5563-6.  Google Scholar

[8]

D. M. Stipanović, Sriram and C. J. Tomlin, Strategies for agents in multi-player pursuit-evasion games, in "Proceedings of the Eleventh International Symposium on Dynamic Games and Applications," Tucson, Arizona, 2004. Google Scholar

[9]

D. M. Stipanović, P. F. Hokayem, M. W. Spong and D. D. Šiljak, Cooperative avoidance control for multi-agent systems, Journal of Dynamic Systems, Measurement and Control, 129 (2007), 699-707. Google Scholar

[10]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations, Annals of Dynamic Games, 10 (2009), 133-145. Google Scholar

[11]

D. M. Stipanović, A. Melikyan and N. Hovakimyan, Guaranteed strategies for nonlinear multi-player pursuit-evasion games, International Game Theory Review, 12 (2010), 1-17. Google Scholar

[12]

D. M. Stipanović, C. J. Tomlin and G. Leitmann, Monotone approximations of minimum and maximum functions and multi-objective problems,, Submitted to Applied Mathematics & Optimization., ().   Google Scholar

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