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Cognitive Economics pp 245–266Cite as

Elements of Viability Theory for the Analysis of Dynamic Economics

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Abstract

The main purpose of viability theory is to explain the evolution of the state of a control system, governed by nondeterministic dynamics and subjected to viability constraints, to reveal the concealed feedbacks which allow the system to be regulated and provide selection mechanisms for implementing them.

It assumes implicitly an “opportunistic” and “conservative” behavior of the system: a behavior which enables the system to keep viable solutions as long as its potential for exploration (or its lack of determinism) — described by the availability of several evolutions — makes possible its regulation.

Examples of viability constraints are provided by architectures of networks imposing constraints described by connectionist tensors operating on coalitions of actors linked by the network. The question raised is how to modify a given dynamic system governing the evolution of the signals, the connectionist tensors and the coalitions in such a way that the architecture remains viable.

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References

  1. Aubin J.-P. (1991) Viability Theory (second edition in preparation) Birkhäuser, Boston, Basel, Berlin

    Google Scholar 

  2. Aubin J.-P. (1996) Neural Networks and Qualitative Physics: A Viability Approach,Cambridge University Press

    Google Scholar 

  3. Aubin J.-P. (1997) Dynamic Economic Theory: A Viability Approach,Springer-Verlag (second edition in preparation)

    Google Scholar 

  4. Aubin J.-P. (1998) Minimal Complexity and Maximal Decentralization, in KNOWLEDGE AND INFORMATION IN A DYNAMIC ECONOMY, Beckmann H.J., Johansson B., Snickars F & Thord D. Eds, Springer, 83–104

    Google Scholar 

  5. Aubin J.-P. (1998) Optima and Equilibria,Springer-Verlag (second edition)

    Google Scholar 

  6. Aubin J.-P. (1999) Mutational and morphological analysis: tools for shape regulation and morphogenesis,Birkhäuser

    Google Scholar 

  7. Aubin J.-P. (1999) Impulse Differential Inclusions and Hybrid Systems: A Viability Approach, Lecture Notes, University of California at Berkeley

    Google Scholar 

  8. Aubin J.-P. (2000) Applied functional analysis,Wiley-Interscience 265

    Google Scholar 

  9. Aubin J.-P. (2001) Viability Kernels and Capture Basins of Sets under Differential Inclusions, SIAM J. Control, 40, 853–881

    Google Scholar 

  10. Aubin J.-P. (2001) A Concise Introduction to Viability Theory, Optimal Control and Robotics, cours DEA MVA, Ecole Normale Supérieure de Cachan

    Google Scholar 

  11. Aubin J.-P. (2001) Regulation of the Evolution of the Architecture of a Network by Connectionist Tensors Operating on Coalitions of Actors, preprint

    Google Scholar 

  12. Aubin J.-P. (2002) Dynamic Core of Fuzzy Dynamical Cooperative Games, Annals of Dynamic Games, Ninth International Symposium on Dynamical Games and Applications, Adelaide, 2000

    Google Scholar 

  13. Aubin J.-P. (2002) Boundary-Value Problems for Systems of Hamilton-JacobiBellman Inclusions with Constraints, SIAM J. Control

    Google Scholar 

  14. Aubin J.-P. & Burnod Y. (1998) Hebbian Learning in Neural Networks with Gates, Cahiers du Centre de Recherche Viabilité, Jeux, Contrôle # 981

    Google Scholar 

  15. Aubin J.-P. & Cellina A. (1984) Differential inclusions,Springer-Velag

    Google Scholar 

  16. Aubin J.-P. & Da Prato G. (1998) The viability theorem for stochastic differential inclusions, Stochastic Analysis and Applications, 16, 1–15

    Article  Google Scholar 

  17. Aubin J.-P. & Dordan O. (1996) Fuzzy Systems, Viability Theory and Toll Sets, In Handbook of Fuzzy Systems, Modeling and Control, Hung Nguyen Ed.. Kluwer, 461–488

    Google Scholar 

  18. Aubin J.-P. & Dordan O. (2001) Dynamical Qualitative Analysis of Evolutionary Systems, Proceedings of the ECC 2001 Conference

    Google Scholar 

  19. Aubin J.-P. & Doss H. (to appear) Characterization of Stochastic Viability of any Nonsmooth Set Involving its Generalized Contingent Curvature, Stochastic Analysis and Applications

    Google Scholar 

  20. Aubin J.-P. & Frankowska H. (1990) Set-Valued Analysis,Birkhäuser, (second edition in preparation)

    Google Scholar 

  21. Aubin J.-P. & Haddad G. (1991) Cadenced runs of impulse and hybrid control systems, International Journal Robust and Nonlinear Control

    Google Scholar 

  22. Aubin J.-P. & Haddad G. (2001) Detectability under Impulse Differential Inclusions, Proceedings of the ECC 2001 Conference

    Google Scholar 

  23. Aubin J.-P., Lygeros J., Quincampoix. M., Sastry S. & Seube N. (2002) Impulse Differential Inclusions: A Viability Approach to Hybrid Systems, IEEE Transactions on Automatic Control, 47, 2–20

    Google Scholar 

  24. Aubin J.-P., Pujal D. & Saint-Pierre P. (2001) Dynamic Management of Portfolios with Transaction Costs under Contingent Uncertainty, preprin Scuola Normale di Pisa

    Google Scholar 

  25. Cardaliaguet P. (1994) Domaines dicriminants en jeux différentiels, Thèse de lUniversité de Paris-Dauphine

    Google Scholar 

  26. Cardaliaguet P., Quincampoix M. & Saint-Pierre P. (1999) Set-valued numerical methods for optimal control and differential games, In Stochastic and dif f erential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, 177–247 Birkhäuser

    Google Scholar 

  27. Dordan O. (1995) Analyse qualitative ,Masson

    Google Scholar 

  28. Doyen L. & Saint-Pierre P. (1997) Scale of viability and minimal time of crisis, Set-Valued Analysis, 5, 227–246

    Google Scholar 

  29. Eldredge N. & Gould S.J. (1972) Punctuated equilibria: an alternative to phyletic gradualism, in Models in paleobiology, Schopf T.J.M Ed., Freeman, Cooper & Company, 82–115

    Google Scholar 

  30. Frankowska H. (1989) Optimal trajectories associated to a solution of tangent Hamilton-Jacobi equations, Applied Mathematics and Optimization, 19, 291–311

    Google Scholar 

  31. Frankowska H. (1989) Hamilton-Jacobi equation: viscosity solutions and generalized gradients, J. of Math. Analysis and Appl. 141, 21–26

    Google Scholar 

  32. Frankowska H. (1991) Lower semicontinuous solutions to Hamilton-JacobiBellman equations, Proceedings of 30th CDC Conference, IEEE, Brighton, December 11–13

    Google Scholar 

  33. Frankowska H. (1993) Lower semicontinuous solutions of Hamilton-JacobiBellman equation, SIAM J. on Control and Optimization

    Google Scholar 

  34. Guzzi R. (to appear) La viabilité comme reformulation des processus d’équilibrage des marchés, Revue d’Économie Industrielle

    Google Scholar 

  35. Haddad G. (1981) Monotone viable trajectories for functional differential inclusions, J. Diff. Eq., 42, 1–24

    Google Scholar 

  36. Meyer P.-A. (1996) Artificial life and the animat approach to Artificial Intelligence, in Artificial Intelligence, Boden, (Ed.), Academic Press„ 325–354

    Google Scholar 

  37. Quincampoix M. (1990) Frontières de domaines d’invariance et de viabilité pour des inclusions différentielles avec contraintes, Comptes-Rendus de l’Académie des Sciences, Paris, 311, 411–416

    Google Scholar 

  38. Quincampoix M. & Saint-Pierre P. (1995) An algorithm for viability kernels in Hölderian case: Approximation by discrete viability kernels, J. Math. Syst. Est. and Control, Summary: 115–120

    Google Scholar 

  39. Quincampoix M. & Saint-Pierre P. (1998) An algorithm for viability kernels in Hölderian case: Approximation by discrete viability kernels, J. Math. Syst. Est. and Control, 8, 17–29

    Google Scholar 

  40. Rockafellar R.T. & Wets R. (1997) Variational Analysis, Springer-Verlag

    Google Scholar 

  41. Saint-Pierre P. (1994) Approximation of the viability kernel,Applied Mathematics Si Optimisation, 29, 187–209

    Google Scholar 

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Author information

Authors and Affiliations

  1. Université Paris-Dauphine and CREA, Ecole Polytechnique, Paris, France

    Jean-Pierre Aubin

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  1. Jean-Pierre Aubin
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Editors and Affiliations

  1. École Polytechnique, CNRS CREA, 1, rue Descartes, 75005, Paris, France

    Paul Bourgine

  2. École Normale Supérieure, LPS (CNRS UMR8550) and CENECC, 24, rue Lhomond, 75231, Paris CEDEX 05, France

    Jean-Pierre Nadal

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Aubin, JP. (2004). Elements of Viability Theory for the Analysis of Dynamic Economics. In: Bourgine, P., Nadal, JP. (eds) Cognitive Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24708-1_15

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  • DOI: https://doi.org/10.1007/978-3-540-24708-1_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07336-6

  • Online ISBN: 978-3-540-24708-1

  • eBook Packages: Springer Book Archive

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