# 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
##### References:

show all references

##### References:
 [1] Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51 [2] Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 [3] Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012 [4] Giovanni Colombo, Thuy T. T. Le. Higher order discrete controllability and the approximation of the minimum time function. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4293-4322. doi: 10.3934/dcds.2015.35.4293 [5] Davide La Torre, Simone Marsiglio, Franklin Mendivil, Fabio Privileggi. Public debt dynamics under ambiguity by means of iterated function systems on density functions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (11) : 5873-5903. doi: 10.3934/dcdsb.2021070 [6] Bai-Ni Guo, Feng Qi. Properties and applications of a function involving exponential functions. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1231-1249. doi: 10.3934/cpaa.2009.8.1231 [7] Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475 [8] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3337-3349. doi: 10.3934/dcdss.2020443 [9] Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345 [10] Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007 [11] Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029 [12] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [13] Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008 [14] Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403 [15] Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060 [16] Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019 [17] Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235 [18] Harry L. Johnson, David Russell. Transfer function approach to output specification in certain linear distributed parameter systems. Conference Publications, 2003, 2003 (Special) : 449-458. doi: 10.3934/proc.2003.2003.449 [19] Pablo G. Barrientos, Abbas Fakhari, Aliasghar Sarizadeh. Density of fiberwise orbits in minimal iterated function systems on the circle. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3341-3352. doi: 10.3934/dcds.2014.34.3341 [20] Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597

Impact Factor: