Optimal Control in Space and Time and the Management of Environmental Resources

We present methods and tools that can be used to study dynamic environmental resource management in a spatial setting, to explore spatially dependent regulation, and to understand pattern formation. In particular, we present the maximum principle and its use in the context of the emerging frontier of applications of optimal control of diffusive transport processes to environmental and resource economics. We show how optimal spatiotemporal control induces pattern formation and how deep uncertainty with a spatial structure can be handled with spatial robust control methods. Finally, we show how models with diffusive transport can be extended to allow for long-range effects and more general transport mechanisms.

[1]  Bernt Øksendal,et al.  Optimal Control of Stochastic Partial Differential Equations , 2005 .

[2]  David Zilberman,et al.  The Dynamics of Spatial Pollution - The Case of Phosphorus Runoff from Agricultural Land , 2000 .

[3]  M. Petit.  Dynamic optimization. The calculus of variations and optimal control in economics and management : by Morton I. Kamien and Nancy L. Schwartz. Second Edition. North-Holland (Advanced Textbooks in Economics), Amsterdam and New York, 1991. Pp. xvii+377. ISBN0-444- 01609-0 , 1994 .

[4]  Jose A. Scheinkman,et al.  Duality theory for dynamic optimization models of economics: The continuous time case☆ , 1982 .

[5]  Antonios Armaou,et al.  Robust control of parabolic PDE systems with time-dependent spatial domains , 2001, Autom..

[6]  Carmen Camacho,et al.  On the dynamics of capital accumulation across space , 2008, Eur. J. Oper. Res..

[7]  Raouf Boucekkine,et al.  Spatial dynamics and convergence: The spatial AK model , 2013, J. Econ. Theory.

[8]  A. Yannacopoulos.  Rational expectations models: An approach using forward-backward stochastic differential equations , 2008 .

[9]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[10]  Anastasios Xepapadeas,et al.  Robust Control and Hot Spots in Spatiotemporal Economic Systems , 2014, Dyn. Games Appl..

[11]  D. Zilberman,et al.  Spatially and Intertemporally Efficient Management of Waterlogging , 2004 .

[12]  Alexei Onatski,et al.  Modeling Model Uncertainty , 2002 .

[13]  T. Sargent,et al.  Robust Control and Model Uncertainty , 2001 .

[14]  N. El‐Farra,et al.  Integrating robustness, optimality and constraints in control of nonlinear processes , 2001 .

[15]  H. Prins,et al.  VEGETATION PATTERN FORMATION IN SEMI-ARID GRAZING SYSTEMS , 2001 .

[16]  Anastasios Xepapadeas,et al.  Pattern Formation, Spatial Externalities and Regulation in Coupled Economic-Ecological Systems , 2008 .

[17]  A. Xepapadeas,et al.  The Bioeconomics of Migration: A Selective Review Towards a Modelling Perspective , 2014 .

[18]  Lars Peter Hansen,et al.  Robust control of forward-looking models , 2003 .

[19]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[20]  W. Brock,et al.  Energy balance climate models and general equilibrium optimal mitigation policies , 2013 .

[21]  Lars Peter Hansen,et al.  Robust control and model misspecification , 2006, J. Econ. Theory.

[22]  James E. Wilen,et al.  Economics of Spatial-Dynamic Processes , 2007 .

[23]  K. Judd.  Numerical methods in economics , 1998 .

[24]  Andrew J. Weaver,et al.  An atmospheric energy-moisture balance model: Climatology, interpentadal climate change, and coupling to an ocean general circulation model , 1996 .

[25]  Carmen Camacho,et al.  The Spatial Solow Model , 2004 .

[26]  Anastasios Xepapadeas,et al.  Environmental policy, first nature advantage and the emergence of economic clusters , 2013 .

[27]  T. Sargent,et al.  Acknowledging Misspecification in Macroeconomic Theory , 2001 .

[28]  Anastasios Xepapadeas,et al.  Robust Control in Water Management , 2004 .

[29]  G. Thompson,et al.  Necessary and sufficient conditions for optimal control of quasilinear partial differential systems , 1984 .

[30]  The economics of land‐use regulation in the presence of an externality: a dynamic approach , 2007 .

[31]  W. Brock,et al.  Spatial Climate-Economic Models in the Design of Optimal Climate Policies Across Locations , 2012 .

[32]  Marika M. Holland,et al.  The UVic earth system climate model: Model description, climatology, and applications to past, present and future climates , 2001, Data, Models and Analysis.

[33]  Raouf Boucekkine,et al.  On the Optimal Control of Some Parabolic Partial Differential Equations Arising in Economics , 2013 .

[34]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[35]  Raouf Boucekkine,et al.  BRIDGING THE GAP BETWEEN GROWTH THEORY AND THE NEW ECONOMIC GEOGRAPHY: THE SPATIAL RAMSEY MODEL , 2009, Macroeconomic Dynamics.

[36]  Suresh P. Sethi,et al.  Distributed Parameter Systems Approach to the Optimal Cattle Ranching Problem , 1980 .

[37]  K. Desmet,et al.  On Spatial Dynamics , 2010 .

[38]  Roland Glowinski,et al.  Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach , 2008 .

[39]  James N. Sanchirico,et al.  The Economics of Spatial-Dynamic Processes: Applications to Renewable Resources , 2007 .

[40]  Enrique Zuazua,et al.  Controllability and Observability of Partial Differential Equations: Some Results and Open Problems , 2007 .

[41]  Per Krusell,et al.  Economics and Climate Change: Integrated Assessment in a Multi-Region World , 2012 .

[42]  J. R. Bates,et al.  Polar amplification of surface warming on an aquaplanet in “ghost forcing” experiments without sea ice feedbacks , 2005 .

[43]  M. Magill,et al.  Some new results on the local stability of the process of capital accumulation , 1977 .

[44]  M. Magill.  A local analysis of N-sector capital accumulation under uncertainty , 1977 .

[45]  Roland Glowinski,et al.  Exact and Approximate Controllability for Distributed Parameter Systems: Index of subjects , 2008 .

[46]  Anastasios Xepapadeas,et al.  Robust Control of a Spatially Distributed Commercial Fishery , 2013 .

[47]  Takao Asano,et al.  Precautionary Principle and the Optimal Timing of Environmental Policy Under Ambiguity , 2010 .

[48]  Pascal J. Maenhout.  Robust Portfolio Rules and Asset Pricing , 2004 .

[49]  Y. Papageorgiou,et al.  AGGLOMERATION AS LOCAL INSTABILITY OF SPATIALLY UNIFORM STEADY-STATES , 1983 .

[50]  Anastasios Xepapadeas,et al.  Diffusion-Induced Instability and Pattern Formation in Infinite Horizon Recursive Optimal Control , 2006 .

[51]  Kai Leitemo,et al.  Robust Monetary Policy in the New-Keynesian Framework , 2004 .

[52]  William D. Nordhaus,et al.  Economic aspects of global warming in a post-Copenhagen environment , 2010, Proceedings of the National Academy of Sciences.

[53]  A. Xepapadeas,et al.  Model Uncertainty, Ambiguity and the Precautionary Principle: Implications for Biodiversity Management , 2008 .

[54]  Abraham Wald,et al.  Statistical Decision Functions , 1951 .

[55]  Paulo Brito,et al.  Global Endogenous Growth and Distributional Dynamics , 2011 .

[56]  Vilmos Komornik,et al.  Fourier Series in Control Theory , 2005 .

[57]  David Zilberman,et al.  Optimal dynamic pricing of water in the presence of waterlogging and spatial heterogeneity of land , 2004 .

[58]  V. Alexeev,et al.  Polar amplification: is atmospheric heat transport important? , 2013, Climate Dynamics.

[59]  W. Brock,et al.  Spatial externalities and agglomeration in a competitive industry , 2014 .

[60]  Pascal J. Maenhout.  Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium , 2006, J. Econ. Theory.

[61]  Anastasios Xepapadeas,et al.  Pollution Control with Uncertain Stock Dynamics: When, and How, to be Precautious , 2012 .