Optimal control of a crop irrigation model under water scarcity. (English) Zbl 1486.92342
Summary: We consider a simple crop irrigation model and study the optimal control which consists of maximizing the biomass production at harvesting time. A specificity of this work is to impose a quota on the water used for irrigation, in a context of limited resources. The model is written as a 2d non-autonomous dynamical system with a state constraint, and a non-smooth right member given by threshold-based soil and crop water stress functions. We show that when the water quota is below the threshold giving the largest possible production, the optimal strategy consists of irrigating once. We then show that the optimal solution can have one or several singular arcs, and therefore be better than simple bang-bang controls, as commonly used. The gains over the best bang-bang controls are illustrated on numerical simulations. These new feedback controls that we obtain are a promising first step towards the concrete application of control theory to the problem of optimal irrigation scheduling under water scarcity.
MSC:
| 92F05 | Other natural sciences (mathematical treatment) |
| 49N90 | Applications of optimal control and differential games |
| 49N35 | Optimal feedback synthesis |
References:
| [1] | KlossS, PushpalathaR, KamoyoKJ, SchützeN. Evaluation of crop models for simulating and optimizing deficit irrigation systems in arid and semi‐arid countries under climate variability. Water Resour Manag. 2012;26(4):997‐1014. |
| [2] | LiZ, CournèdePH, MailholJC. Irrigation optimization by modeling of plant‐soil interaction. Paper presented at: Proceedings of the 2011 Conference on Applied Simulation and Modelling. Crete, Greece; June 22 ‐ 24, 2011. |
| [3] | SchützeN, SchmitzGH. New planning tool for optimal climate change adaption strategies in irrigation. J Irrig Drain Eng. 2010;136(12):836‐846. |
| [4] | SchützeN, dePalyM, ShamirU. Novel simulation‐based algorithms for optimal open‐loop and closed‐loop scheduling of deficit irrigation systems. J Hydroinf. 2011;14(1):136‐151. |
| [5] | VicoG, PorporatoA. Probabilistic description of crop development and irrigation water requirements with stochastic rainfall. Water Resour Res. 2013;49(3):1466‐1482. |
| [6] | ChevironB, VervoortRW, AlbashaR, DaironR, Le PriolC, MailholJ. A framework to use crop models for multi‐objective constrained optimization of irrigation strategies. Environ Model Softw. 2016;86:145‐157. |
| [7] | LopesSO, FontesFACC, PereiraRMS, dePinhoMR, GonçalvesAM. Optimal control applied to an irrigation planning problem. Math Probl Eng. 2016:5076879. · Zbl 1400.49048 |
| [8] | ShaniU, TsurY, ZemelA. Optimal dynamics irrigation schemes. Opt Control Appl Methods. 2004;25:91‐106. · Zbl 1073.91058 |
| [9] | KalboussiN, RouxS, BoumazaK, SinfortC, RapaportA. About modeling and control strategies for scheduling crop irrigation. Paper presented at: Proceedings of the IFAC Workshop on Control Methods for Water Resource Systems CMWRS, Delft (Netherlands), IFAC‐PapersOnLine. Crete, Greece; Vol. 52, 2019:43‐48. June 22 ‐ 24, 2011. |
| [10] | PelakN, RevelliaR, PorporatoA. A dynamical systems framework for crop models: toward optimal fertilization and irrigation strategies under climatic variability. Ecol Model. 2017;365:80‐92. |
| [11] | BertrandN, RouxS, ForeyO, GuinetM, WeryJ. Simulating plant water stress dynamics in a wide range of bi‐specific agrosystems in a region using the BISWAT model. Eur J Agron. 2018;99:116‐128. |
| [12] | StedutoP, HsiaoH, RaesD, FereresE. AquaCrop, the FAO crop model to simulate yield response to water: I concepts and underlying principles. Agronomy J. 2009;101(3):426‐437. |
| [13] | AllenR, PereiraL, RaesD, SmithM, et al. Crop Evapotranspiration ‐ Guidelines for Computing Crop Water Requirements ‐ FAO Irrigation and Drainage Paper 56. FAO, Roma (Italy); 1998. |
| [14] | VinterR. Optimal Control, Systems & Control: Foundations & Applications. Boston: Birkhäuser; 2000. · Zbl 0952.49001 |
| [15] | MieleA. In: LeitmannG (ed.), ed. Extremization of Linear Integrals by Green’s Theorem, Optimization Techniques. New York, NY: Academic Press; 1962:69‐98. |
| [16] | HermesH, LasalleJP. Functional Analysis and Time Optimal Control. Mathematics in Science and Engineering. Vol 56. New York: Elsevier; 1969. · Zbl 0203.47504 |
| [17] | HartlRF, FeichtingerG. A new sufficient condition for most rapid approach paths. J Optim Theory Appl Volucella. 1987;54(2):402‐411. · Zbl 0597.49015 |
| [18] | RapaportR, CartignyP. Turnpike theorems by a value function approach. ESAIM Control Optim Calculus Variat. 2004;10:123‐141. · Zbl 1068.49016 |
| [19] | TrélatE, ZuazuaE. The turnpike property in finite‐dimensional nonlinear optimal control. J Differ Equ. 2015;258:81‐114. · Zbl 1301.49010 |
| [20] | FaulwasserT, KordaM, JonesCN, BonvinD. On turnpike and disspativity properties of continuous‐time optimal control problems. Automatica. 2017;81:297‐304. · Zbl 1373.49026 |
| [21] | SpenceM, StarrettD. Most rapid approach paths in accumulation problems. Int Econ Rev. 1975;16:388‐403. · Zbl 0316.90009 |
| [22] | WalterW. Ordinary Differential Equations. New York, NY: Springer; 1998. · Zbl 0991.34001 |
| [23] | ClarkeF. Functional Analysis, Calculus of Variations and Optimal Control. New York, NY: Springer; 2013. · Zbl 1277.49001 |
| [24] | BonnansF, GiorgiD, HeymannB, MartinonP, TissotO. Bocop HJB 1.0.1 ‐ User Guide. Technical Report INRIA 0467; 2015. |
| [25] | BrumbelowK, GeorgakakosA. Determining crop‐water production functions using yield‐irrigation gradient algorithms. Agric Water Manag. 2007;87:151‐161. |
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