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

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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

Abstract
Related Papers

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 and 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," 2^nd edition, Springer-Verlag, Berlin, Germany, 2005.
[2]	R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques," Birkhäuser, Boston, MA, 1996.
[3]	A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover Publications, New York, NY, 1975.
[4]	G. Leitmann and J. Skowronski, Avoidance control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298.
[5]	G. Leitmann, Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, 32 (1980), 569-576. doi: 10.1007/BF00934040.
[6]	G. Leitmann and J. Skowronski, A note on avoidance control, Optimal Control Applications & Methods, 4 (1983), 335-342. doi: 10.1002/oca.4660040406.
[7]	K. M. Miettinen, "Nonlinear Multiobjective Optimization," Kluwer Academic Publishers, Norwell, MA, 1999. doi: 10.1007/978-1-4615-5563-6.
[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.
[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.
[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.
[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.
[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.

show all references

References:

[1]	A. Bacciotti and L. Rosier, "Liapunov Functions and Stability in Control Theory," 2^nd edition, Springer-Verlag, Berlin, Germany, 2005.
[2]	R. A. Freeman and P. V. Kokotović, "Robust Nonlinear Control Design: State Space and Lyapunov Techniques," Birkhäuser, Boston, MA, 1996.
[3]	A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Dover Publications, New York, NY, 1975.
[4]	G. Leitmann and J. Skowronski, Avoidance control, Journal of Optimization Theory and Applications, 23 (1977), 581-591. doi: 10.1007/BF00933298.
[5]	G. Leitmann, Guaranteed avoidance strategies, Journal of Optimization Theory and Applications, 32 (1980), 569-576. doi: 10.1007/BF00934040.
[6]	G. Leitmann and J. Skowronski, A note on avoidance control, Optimal Control Applications & Methods, 4 (1983), 335-342. doi: 10.1002/oca.4660040406.
[7]	K. M. Miettinen, "Nonlinear Multiobjective Optimization," Kluwer Academic Publishers, Norwell, MA, 1999. doi: 10.1007/978-1-4615-5563-6.
[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.
[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.
[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.
[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.
[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.

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