Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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The value function of the shallow lake problem as a viscosity solution of a HJB equation


Authors: Georgios Kossioris and Christos Zohios
Journal: Quart. Appl. Math. 70 (2012), 625-657
MSC (2010): Primary 49L25
DOI: https://doi.org/10.1090/S0033-569X-2012-01253-2
Published electronically: June 21, 2012
MathSciNet review: 3052082
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Abstract | References | Similar Articles | Additional Information

Abstract: The economic analysis of a shallow lake ecological system requires the study of a nonstandard optimal control problem due to the conflicting services it provides and the nonlinearity of the governing dynamics. We first investigate the geometry of the optimal control-optimal path pair, by standard control analysis, for a given range of the discount factor. We then consider the welfare function (value function) as a viscosity solution of a reduced Hamilton-Jacobi-Bellman equation and we prove various regularity properties which are related to the dynamics of the problem. Finally, we approximate the welfare function by monotone convergent numerical schemes and present the numerical results.


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Additional Information

Georgios Kossioris
Affiliation: Department of Mathematics, University of Crete, Knossou Avenue, P.O. BOX 2208, Heraklion 71409, Greece
Email: kosioris@math.uoc.gr

Christos Zohios
Affiliation: Department of Mathematics, University of Crete, Knossou Avenue, P.O. BOX 2208, Heraklion 71409, Greece
Email: zohios@math.uoc.gr

DOI: https://doi.org/10.1090/S0033-569X-2012-01253-2
Received by editor(s): September 20, 2010
Published electronically: June 21, 2012
Additional Notes: The authors would like to thank Prof. A. Xepapadeas and Prof. G. Zouraris for stimulating discussions and suggestions
Article copyright: © Copyright 2012 Brown University

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