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 Mat. Sb., 2007, Volume 198, Number 1, Pages 21–42 (Mi msb1438)

Generic phase transitions and profit singularities in Arnol'd's model

A. A. Davydovab*, H. Mena Matosc

b International Institute for Applied Systems Analysis
c University of Porto

Abstract: For a smooth one-parameter family of pairs of control systems and profit densities on a circle, the generic transitions between optimal rotations and stationary strategies are studied in the problem of maximization of the time-averaged profit on the infinite horizon. It is shown that there are only two types of such transitions, the corresponding singularities of the average profit as a function of the family parameter are found, and it is proved that these singularities are stable under small perturbations of a generic family. The classification of singularities of the maximum average profit is completed for generic families.
Bibliography: 16 titles.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm1438

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English version:
Sbornik: Mathematics, 2007, 198:1, 17–37

Bibliographic databases:

UDC: 517.9
MSC: Primary 49J15; Secondary 49J45, 49N20, 49N60, 58K25, 58K40, 90C31

Citation: A. A. Davydov, H. Mena Matos, “Generic phase transitions and profit singularities in Arnol'd's model”, Mat. Sb., 198:1 (2007), 21–42; Sb. Math., 198:1 (2007), 17–37

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1438
• https://doi.org/10.4213/sm1438
• http://mi.mathnet.ru/eng/msb/v198/i1/p21

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Davydov AA, Kukshina EO, “Typical profit singularities of one-parametric cyclic process with fixed period”, Optimization, 57:2 (2008), 205–214
2. Mena-Matos H., “Generic profit singularities in time averaged optimization. The case of a control space with a regular boundary”, J. Dyn. Control Syst., 16:1 (2010), 101–120
3. A. A. Davydov, T. S. Shutkina, “Uniqueness of a cycle with discounting that is optimal with respect to the average time profit”, Proc. Steklov Inst. Math. (Suppl.), 276, suppl. 1 (2012), S80–S87
4. A. A. Davydov, H. Mena-Matos, C. S. Moreira, “Generic profit singularities in time-averaged optimization for cyclic processes in polydynamical systems”, Journal of Mathematical Sciences, 199:5 (2014), 510–534
5. A.A. Davydov, H. Mena-Matos, C.S. Moreira, “Generic profit singularities in time averaged optimization for phase transitions in polydynamical systems”, Journal of Mathematical Analysis and Applications, 2014
6. A. Belyakov, A. A. Davydov, “Efficiency optimization for the cyclic use of a renewable resource”, Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 14–21
7. Nikolay A. Krasovskii, Alexander M. Tarasyev, “Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals”, Autom. Remote Control, 79:6 (2018), 1148–1167
8. A. A. Davydov, “Existence of Optimal Stationary States of Exploited Populations with Diffusion”, Proc. Steklov Inst. Math., 310 (2020), 124–130
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