# American Institute of Mathematical Sciences

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

 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].
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
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