Applied Mathematics & Optimization

, Volume 66, Issue 3, pp 455–473

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

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

DOI: 10.1007/s00245-012-9179-8

Cite this article as:
Stipanović, D.M., Tomlin, C.J. & Leitmann, G. Appl Math Optim (2012) 66: 455. doi:10.1007/s00245-012-9179-8

Abstract

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.

Keywords

Dynamic systems Multiple objectives Minimum function Maximum function Approximations of functions 

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