An optimal control framework to support the management and control of resources in a wide range of problems arising in agriculture is discussed. Lessons extracted from past research on the weed control problem and a survey of a vast body of pertinent literature led to the specification of key requirements to be met by a suitable optimization framework. The proposed layered control structure—including planning, coordination, and execution layers—relies on a set of nested optimization processes of which an “infinite horizon” Model Predictive Control scheme plays a key role in planning and coordination. Some challenges and recent results on the Pontryagin Maximum Principle for infinite horizon optimal control are also discussed. 1. Introduction This paper concerns a framework based on dynamic optimization and, in particular, optimal control to support the management and control of resources in a wide range of problems arising in agriculture. The relevance of optimizing resources in agriculture is undeniable. In fact, the need to feed a growing population on earth while keeping a sustainable equilibrium with the environment is one of the most critical challenges that humankind is facing [1, 2]. Being a fact that this general issue concerns the general problem of food production resources and of global resources management of the earth, and thus, transcends the strict context of agriculture [1–6], it is absolutely clear that the scarcity of resources and the multiple, mostly adverse, impacts in the environment call for advanced management practices in which optimization should be increasingly pervasive and effective [7, 8]. It is not surprising that the research community has been enthusiastically embracing the huge challenges arising in the extremely wide range of research and management problems arising in this context, as it is attested by a vast amount of technical literature. These challenges span the whole spectrum of relevant issues: from the optimization problem formulation—encompassing the specification of the scope of the interactions to be considered, the modeling of system's dynamics, the problem constraints and performance criteria, and the optimization time horizon—to the approaches to solve it—which may draw from optimization and control theory results, notably the characterization of solutions to their approximation, analysis of sensitivity and robustness of the solution, and the numerical issues arising in the computation procedures—, and to the framework to integrate the generated output in appropriate decision-making and control support
References
[1]
Y. Farzin, “Sustainability, optimality, and development policy,” Tech. Rep., Department of Agricultural and Resource Economics, University of California, Davis, Calif, USA, 2008.
[2]
Y. Mariam, J. Galaty, and G. Coffin, “Strategic decision-making: adoption of agricultural technologies and risk in a peasant economy,” Munich Personal RePEc Archive 387, 2006.
[3]
W. Djezou, “Agriculture and deforestation: what optimal conversion of forest land to agriculture in C？te d’Ivoire?” Tech. Rep., University of Cocody-Abidjan, 2009.
[4]
A. Huhtala and M. Laukkanen, Optimal Control of Dynamic Point and Nonpoint Pollution in a Coastal Ecosystem: Agricultural Abatement Versus Investment in Waste Water Treatment Plants, MTT Agrifood Research, Jokioinen, Finland, 2011.
[5]
J. B. de Sousa, B. Maciel, and F. L. Pereira, “Sensor systems on networked vehicles,” Networks and Heterogeneous Media, vol. 4, no. 2, pp. 223–247, 2009.
[6]
N. S. G. Williams, A. K. Hahs, and J. W. Morgan, “A dispersal-constrained habitat suitability model for predicting invasion of alpine vegetation,” Ecological Applications, vol. 18, no. 2, pp. 347–359, 2008.
[7]
C. Galvan, I. Pacheco, R. Gonzalez et al., “Review. Advantages and disadvantages of control theories applied in greenhouse climate control systems,” Spanish Journal of Agricultural Research, vol. 10, no. 4, pp. 926–938, 2012.
[8]
D. Zilberman, “The use and potential of optimal control models in agricultural economics,” Western Journal of Agricultural Economics, vol. 7, no. 2, pp. 395–406, 1982.
[9]
F. Pereira and G. Silva, “Necessary conditions of optimality for state constrained infinite horizon differential inclusions,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pp. 6717–6722, Orlando, Fla, USA, December 2011.
[10]
E. Stiegelmeier, Modelo de otimiza？？o para o controle de plantas daninhas usando programa？？o n？o linear inteira mista [Ph.D. thesis], USP, S？o Paulo, Brazil, 2012.
[11]
E. Stiegelmeier, V. Oliveira, G. Silva, D. Karam, M. Furlan, and H. Kajino, “Herbicide application optimization model for weed control using the resistance dynamics,” Tech. Rep., University of S？o Paulo, S？o Carlos, Brazil, 2012.
[12]
F. A. C. C. Fontes, “A general framework to design stabilizing nonlinear model predictive controllers,” Systems and Control Letters, vol. 42, no. 2, pp. 127–143, 2001.
[13]
F. Fontes, L. Magni, and E. Gyurkovics, “Sampled-data model predictive control for nonlinear time-varying systems: stability and robustness,” in Assessment and Future Directions of Nonlinear Model Predictive Control, F. Allgower, R. Findeisen, and L. Biegler, Eds., vol. 358 of Lecture Notes in Control and Information Systems, pp. 115–129, Springer, Heidelberg, Germany, 2007.
[14]
S. L. Fraga and F. L. Pereira, “Hamilton-Jacobi-Bellman equation and feedback synthesis for impulsive control,” IEEE Transactions on Automatic Control, vol. 57, no. 1, pp. 244–249, 2012.
[15]
F. Fontes and F. Pereira, “Model predictive control of impulsive dynamical systems,” in Proceedings of the 4th IFAC Nonlinear Model Predictive Control Conference (NMPC '12), Noordwijkerhout, The Netherlands, August 2012.
[16]
S. O. Lopes, F. A. C. C. Fontes, and M. do Rosário de Pinho, “On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems,” Discrete and Continuous Dynamical Systems, vol. 29, no. 2, pp. 559–575, 2011.
[17]
R. Gorddard, D. Pannell, and G. Hertzler, “An optimal control model for integrated weed management under herbicide resistance,” Australian Journal of Agricultural and Resource Economics, vol. 39, no. 1, pp. 71–87, 1995.
[18]
S. Manalil, R. Busi, M. Renton, and S. B. Powles, “Rapid evolution of herbicide resistance by low herbicide dosages,” Weed Science, vol. 59, no. 2, pp. 210–217, 2011.
[19]
R. Jones and O. Cacho, “A dynamic optimisation model of weed control,” Working Paper Series on Agricultural and Resource Economics, University of New England, 2000.
[20]
T. Kompas and L. Chu, “A rule of thumb for controlling invasive weeds: an application to hawkweed in Australia,” Tech. Rep. 70, Environmental Economics Research Hub Research Reports, Australian National University, 2010.
[21]
M. Wetzstein, P. Szmedra, W. Musser, and C. C. Chou, “Optimal agricultural pest management with multiple species,” Northeastern Journal of Agricultural and Resource Economics, vol. 14, no. 1, pp. 71–77, 1985.
[22]
M. Rafikov and J. M. Balthazar, “Optimal pest control problem in population dynamics,” Computational and Applied Mathematics, vol. 24, no. 1, pp. 65–81, 2005.
[23]
T. Christiaans, T. Eichner, and R. Pethig, “Optimal pest control in agriculture,” Journal of Economic Dynamics and Control, vol. 31, no. 12, pp. 3965–3985, 2007.
[24]
P. Georgescu and G. Moro？anu, “Pest regulation by means of impulsive controls,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 790–803, 2007.
[25]
M. Rafikov, A. Lordelo, and E. Rafikova, “Impulsive biological pest control strategies of the sugarcane borer,” Mathematical Problems in Engineering, vol. 2012, Article ID 726783, 14 pages, 2012.
[26]
B. Liu, Q. Zhang, and Y. Gao, “The dynamics of pest control pollution model with age structure and time delay,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2814–2823, 2010.
[27]
Y. Pei, X. Ji, and C. Li, “Pest regulation by means of continuous and impulsive nonlinear controls,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 810–822, 2010.
[28]
A. Arutyunov, D. Karamzin, and F. L. Pereira, “On a generalization of the impulsive control concept: controlling system jumps,” Discrete and Continuous Dynamical Systems, vol. 29, no. 2, pp. 403–415, 2011.
[29]
F. L. Pereira and G. N. Silva, “Necessary conditions of optimality for vector-valued impulsive control problems,” Systems and Control Letters, vol. 40, no. 3, pp. 205–215, 2000.
[30]
R. Epanchin-Niell and J. Wilen, “Optimal control of spatial-dynamic processes: the case of biological invasions,” Discussion Paper RFF DP 11-07, Resources for the Future, 2011, http://www.rff.org/.
[31]
S. A. Gourley and X. Zou, “A mathematical model for the control and eradication of a wood boring beetle infestation,” SIAM Review, vol. 53, no. 2, pp. 321–345, 2011.
[32]
G. Schamel and S. F. Schubert, “A dynamic optimal control model of crop thinning,” in Proceedings of the International Association of Agricultural Economists (IAAE) Triennial Conference, pp. 18–24, Foz do Lgua？u, Brazil, August 2012.
[33]
D. Fleisher and H. Baruh, An Optimal Control Strategy For Crop Growth in Advanced Life Support Systems, Rutgers University,, New Brunswick, NJ, USA, 2000.
[34]
W. Hediger, “Optimal control of soil erosion and phosphorous runoff from agricultural land,” Agricultural Economics, Swiss Federal Institute of Technology, CH-8092 Zurich, Switzerland, March 2000.
[35]
L. Ragot and K. Schuber, “The optimal carbon sequestration in agricultural soils: does the dynamics of the physical process matter?” Tech. Rep. 2006.40, Centre d’Economie de la Sorbonne UMR 8174, 2006.
[36]
B. Sohngen and R. Mendelsohn, “An optimal control model of forest carbon sequestration,” American Journal of Agricultural Economics, vol. 85, no. 2, pp. 448–457, 2003.
[37]
G. Doole, “Numerical optimisation of multiple-phase systems incorporating transition costs,” Tech. Rep., School of Agricultural and Resource Economics, University of Western Australia, 2006.
[38]
J. Krawczyk and O. Serea, “A viability theory approach to a two stage optimal control problem of technology adoption,” in Proceedings of the Joint Meeting of the American Mathematical Society and the New Zealand Mathematical Society (AMS-NZMS '07), Victoria University of Wellington, Wellington, New Zealand, December 2007.
[39]
R. Qi, Optimization and optimal control of plant growth: application application of greenLab model for decision aid in agriculture [Ph.D. thesis], école Centrale des Arts e Manufactures, école Centrale Paris, 2004.
[40]
K. Tomiyama, “Two-stage optimal control problems and optimality conditions,” Journal of Economic Dynamics and Control, vol. 9, no. 3, pp. 317–337, 1985.
[41]
J. Risbey, M. Kandlikar, H. Dowlatabadi, and D. Graetz, “Scale, context, and decision making in agricultural adaptation to climate variability and change,” in Mitigation and Adaptation Strategies for Global Change, vol. 4, pp. 137–165, Kluwer Academic, New York, NY, USA, 1999.
[42]
C. Fraisse, D. Zierden, N. Breuer, J. Jackson, and C. Brown, “Climate forecast and decision making in agriculture,” Tech. Rep. ABE 352, Institute of Food & Agricultural Sciences (IFAS), University of Florida, 2011.
[43]
J. Russo and J. Zidek, “Operational models for agricultural decision making,” in International Pest Risk Modeling & Mapping Workshop V, Fort Collins Science Center, Fort Collins, Colo, USA, August 2011.
[44]
A. Utset, “Introducing modeling tools to support agricultural decisionmaking under climate change conditions: a Spanish Experience,” Tech. Rep., Instituto Tecnológico Agrário, 2012.
[45]
W. Yi, J. Ping, and L. Yahui, “Study on intelligent decision-making platform in the agricultural production,” in Proceedings of the 4th International Conference on Intelligent Computation Technology and Automation (ICICTA '11), pp. 224–227, Shenzhen, China, March 2011.
[46]
H. Passam, A. Tocatlidou, B. Mahaman, and A. Sideridis, “Methods for decision making with insufficient knowledge in agriculture,” in Proceedings of the Conference of the European Federation for Information Technology in Agriculture, Food, and the Environment (EFITA '03), pp. 727–731, Budapest, Hungary, July 2003.
[47]
S. S. Seefeldt, J. E. Jensen, and E. P. Feurst, “Log-logistic analysis of herbicide dose-response relationships,” Weed Technology, vol. 9, no. 2, pp. 218–227, 1995.
[48]
P. J. Tranel and T. R. Wright, “Resistance of weeds to ALS-inhibiting herbicides: what have we learned?” Weed Science, vol. 50, no. 6, pp. 700–712, 2002.
A. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems, Springer, Dordrecht, The Netherlands, 2000.
[51]
A. V. Arutyunov, D. Y. Karamzin, and F. L. Pereira, “The maximum principle for optimal control problems with state constraints by R.V. gamkrelidze: revisited,” Journal of Optimization Theory and Applications, vol. 149, no. 3, pp. 474–493, 2011.
[52]
F. Clarke, Y. Ledyaev, R. Stern, and P. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, NY, USA, 1998.
[53]
L. Pontryagin, V. Boltyanskiy, R. Gamkrelidze, and E. Mishchenko, Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, NY, USA, 1962.
[54]
R. Vinter, Optimal Control, Birkh？user, Boston, Mass, USA, 2000.
[55]
M. Caputo, “Necessary and sufficient conditions for infinite horizon control problems,” in Foundations of Dynamic Economic Analysis Optimal Control Theory and Applications, pp. 381–411, Cambridge University Press, New York, NY, USA, 2005.
[56]
D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000.
[57]
J. E. da Silva and J. B. de Sousa, “A dynamic programming based path-following controller for autonomous vehicles,” Control and Intelligent Systems, vol. 39, no. 4, pp. 245–253, 2011.
