Applied Mathematics & Optimization

, Volume 66, Issue 3, pp 455–473 | Cite as

Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems

  • Dušan M. Stipanović
  • Claire J. Tomlin
  • George Leitmann


In this paper 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 derived 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 with two evaders and two pursuers is provided.


Dynamic systems Multiple objectives Minimum function Maximum function Approximations of functions 


  1. 1.
    Bacciotti, A., Rosier, L.: Liapunov Functions and Stability in Control Theory, 2nd edn. Springer, Berlin (2005) zbMATHGoogle Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997) zbMATHCrossRefGoogle Scholar
  3. 3.
    Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM, Philadelphia (1999). Revised and updated 2nd edn. zbMATHGoogle Scholar
  4. 4.
    Bhat, S.P., Bernstein, D.S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, Springfield (2007) zbMATHGoogle Scholar
  6. 6.
    Bullen, P.S.: Handbook of Means and Their Inequalities. Kluwer Academic, Dordrecht (2003) zbMATHGoogle Scholar
  7. 7.
    Bullen, P.S., Mitrinović, D.S., Vasić, P.M.: Means and Their Inequalities. Reidel, Dordrecht (1988) zbMATHGoogle Scholar
  8. 8.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955) zbMATHGoogle Scholar
  9. 9.
    Falcone, M., Ferretti, R.: Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. Math. 67, 315–344 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988) Google Scholar
  11. 11.
    Filippov, V.V.: Basic Topological Structures of Ordinary Differential Equations. Kluwer Academic, Dordrecht (1998) zbMATHGoogle Scholar
  12. 12.
    Freeman, R.A., Kokotović, P.V.: Robust Nonlinear Control Design: State Space and Lyapunov Techniques. Birkhäuser, Boston (1996) zbMATHGoogle Scholar
  13. 13.
    Fry, R., McManus, S.: Smooth bump functions and the geometry of Banach spaces. Expo. Math. 20, 143–183 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Halmos, P.R.: Measure Theory. Springer, New York (1974) zbMATHGoogle Scholar
  15. 15.
    Hwang, I., Tomlin, C.J.: Protocol-based conflict resolution for air traffic control. Technical Report, SUDAAR-762, Department of Aeronautics and Astronautics, Stanford University (2002) Google Scholar
  16. 16.
    Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover, New York (1975) Google Scholar
  17. 17.
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities: Theory and Applications. Academic Press, New York (1969) zbMATHGoogle Scholar
  18. 18.
    Leitmann, G.: Deterministic control of uncertain systems. Acta Astronaut. 7, 1457–1461 (1980) zbMATHCrossRefGoogle Scholar
  19. 19.
    Leitmann, G.: Guaranteed avoidance strategies. J. Optim. Theory Appl. 32, 569–576 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Leitmann, G.: The Calculus of Variations and Optimal Control: An Introduction. Plenum, New York (1981) zbMATHGoogle Scholar
  21. 21.
    Leitmann, G., Skowronski, J.: Avoidance control. J. Optim. Theory Appl. 23, 581–591 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Leitmann, G., Skowronski, J.: A note on avoidance control. Optim. Control Appl. Methods 4, 335–342 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic, Norwell (1998) CrossRefGoogle Scholar
  24. 24.
    Mitchell, I., Bayen, A.M., Tomlin, C.J.: A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50, 947–957 (2005) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mitrinović, D.S.: Analytic Inequalities. Springer, Berlin (1970). With P.M. Vasić zbMATHGoogle Scholar
  26. 26.
    Stipanović, D.M., Sriram, Tomlin, C.J.: 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
  27. 27.
    Stipanović, D.M., Hokayem, P.F., Spong, M.W., Šiljak, D.D.: Cooperative avoidance control for multi-agent systems. J. Dyn. Syst. Meas. Control 129, 699–707 (2007) CrossRefGoogle Scholar
  28. 28.
    Stipanović, D.M., Melikyan, A., Hovakimyan, N.: Some sufficient conditions for multi-player pursuit-evasion games with continuous and discrete observations. Annals of Dynamic. Games 10, 133–145 (2009) Google Scholar
  29. 29.
    Stipanović, D.M., Melikyan, A., Hovakimyan, N.: Guaranteed strategies for nonlinear multi-player pursuit-evasion games. Int. Game Theory Rev. 12(1), 1–17 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Vaisbord, E.M., Zhukovskiy, V.I.: Introduction to Multi-Player Differential Games and Their Applications. Gordon & Breach, New York (1988) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Dušan M. Stipanović
    • 1
  • Claire J. Tomlin
    • 2
  • George Leitmann
    • 3
  1. 1.Coordinated Science Laboratory, Department of Industrial and Enterprise Systems EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Electrical Engineering and Computer ScienceUniversity of California at BerkeleyBerkeleyUSA
  3. 3.College of EngineeringUniversity of California at BerkeleyBerkeleyUSA

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