The Lovebirds Problem: Why Solve Hamilton-Jacobi-Bellman Equations Matters in Love Affairs

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• 作者：Thierry Goudon (1)
  Pauline Lafitte (2) (3)
  
  1. Project COFFEE ; INRIA Sophia Antipolis M茅diterran茅e & Labo. J.-A. Dieudonn茅 ; UMR 7351 CNRS-Univ. Nice Sophia Antipolis ; Parc Valrose ; 06108 ; Nice ; France
  2. Laboratoire MAS鈥揈cole Centrale Paris Grande Voie des Vignes ; 92290 ; Chatenay-Malabry ; France
  3. F茅d茅ration de Math茅matiques ; CNRS ; 3487 ; Chatenay-Malabry ; France
  
• 关键词：Dynamical systems ; Computation of stable and unstable manifolds ; Dynamical programming principle ; Hamilton ; Jacobi ; Bellman equations ; 37D10 ; 37M99 ; 49L99
• 刊名：Acta Applicandae Mathematicae
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：136
• 期：1
• 页码：147-165
• 全文大小：957 KB
• 参考文献：1. Barles, G. (1994) Solutions de Viscosit茅 des 脡quations de Hamilton-Jacobi. Springer, Paris
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  3. Crandall, M.G., Evans, L.C., Lions, P.-L. (1984) Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 282: pp. 487-502 dx.doi.org/10.1090/S0002-9947-1984-0732102-X" target="_blank" title="It opens in new window">CrossRef
  4. Evans, L.C. (1998) Partial Differential Equations. AMS, Providence
  5. Ewald, C.-O., McNamara, J., Houston, A. (2007) Parental care as a differential game: a dynamic extension of the Houston鈥揇avies game. Appl. Math. Comput. 190: pp. 1450-1465 dx.doi.org/10.1016/j.amc.2007.02.060" target="_blank" title="It opens in new window">CrossRef
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  8. Lions, P.-L. (1982) Generalized Solutions of Hamilton-Jacobi Equations. Pitman, Boston
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  11. Rey, J.-M. (2010) A mathematical model of sentimental dynamics accounting for marital dissolution. PLoS ONE 5: pp. e9881-03 dx.doi.org/10.1371/journal.pone.0009881" target="_blank" title="It opens in new window">CrossRef
  
• 刊物主题：Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9036

文摘

The lovebirds problem consists in finding the compromise between well-being and efforts that are necessary to sustain a sentimental relationship. According to a modeling introduced by J.-M. Rey, the problem can be described as finding the initial data for a certain dynamical system, guaranteeing that the associated trajectory belongs to the stable manifold. We further discuss this interpretation by means of the Dynamical Programming Principle and the Hamilton-Jacobi-Bellman framework. It allows us to propose an algorithm that determines numerically the solution of the lovebirds problem.