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

Kossioris, Georgios; Zohios, Christos

doi:10.1090/S0033-569X-2012-01253-2

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: 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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