{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:16:36Z","timestamp":1757312196850,"version":"3.37.3"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T00:00:00Z","timestamp":1646956800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T00:00:00Z","timestamp":1646956800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["To CHAIR - POCI-01-0145-FEDER-028247"],"award-info":[{"award-number":["To CHAIR - POCI-01-0145-FEDER-028247"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int. J. Appl. Comput. Math"],"published-print":{"date-parts":[[2022,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we study the problem of daily irrigation of an agricultural crop using optimal control. The dynamics is a model based on field capacity modes, where the state, <jats:italic>x<\/jats:italic>, represents the water in the soil and the control variable, <jats:italic>u<\/jats:italic>, is the flow rate of water from irrigation. The variation of water in the soil depends on the field capacity of the soil, <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_{FC}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>FC<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, weather conditions, losses due to deep percolation and irrigation. Our goal is to minimize the amount of water used for irrigation while keeping the crop in a good state of preservation. To enforce such requirement, the state constraint <jats:inline-formula><jats:alternatives><jats:tex-math>$$x(t) \\ge x_{\\min }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>min<\/mml:mo>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is coupled with the dynamics, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_{\\min }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>min<\/mml:mo>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the hydrological need of the crop. Consequently, the problem under study is a <jats:italic>state constrained<\/jats:italic> optimal control problem. Under some mild assumptions, we consider several basic profiles for the optimal trajectories. Appealing to the Maximum Principle (MP), we characterize analytically the solution and its multipliers for each case. We then illustrate the analytical results running some computational simulations, where the analytical information is used to partially validate the computed solutions. The need to study irrigation strategies is of foremost importance nowadays since 80% of the fresh water used on our planet is used in agriculture.<\/jats:p>","DOI":"10.1007\/s40819-022-01266-9","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T10:02:50Z","timestamp":1646992970000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Analytical Study for Different Extremal State Solutions of an Irrigation Optimal Control Problem with Field Capacity Modes"],"prefix":"10.1007","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4592-3426","authenticated-orcid":false,"given":"Ana P.","family":"Lemos-Pai\u00e3o","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2335-5459","authenticated-orcid":false,"given":"Sofia O.","family":"Lopes","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9276-2317","authenticated-orcid":false,"given":"M. d. R.","family":"de Pinho","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,11]]},"reference":[{"issue":"2","key":"1266_CR1","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s10589-008-9221-6","volume":"47","author":"M Baglietto","year":"2010","unstructured":"Baglietto, M., Cervellera, C., Sanguineti, M., Zoppoli, R.: Management of water resource systems in the presence of uncertainties by nonlinear approximation techniques and deterministic sampling. Comput. Optim. Appl. 47(2), 349\u2013376 (2010)","journal-title":"Comput. Optim. Appl."},{"key":"1266_CR2","doi-asserted-by":"crossref","unstructured":"Boumaza, K., Kalboussi, N., Rapaport, A., Roux, S., Sinfort, C.: Optimal control of a crop irrigation model under water scarcity. Optim. Control Appl. Methods. 42(6), 1612\u20131631 (2021)","DOI":"10.1002\/oca.2749"},{"issue":"6","key":"1266_CR3","first-page":"3","volume":"97","author":"P Brown","year":"2010","unstructured":"Brown, P., Cochrane, T., Krom, T.: Optimal on-farm irrigation scheduling with a seasonal water limit using simulated annealing. Fundam. Prikl. Mat. 97(6), 3\u201319 (2010)","journal-title":"Fundam. Prikl. Mat."},{"key":"1266_CR4","unstructured":"Clarke, F.\u00a0H.: Optimization and nonsmooth analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, (1983). A Wiley-Interscience Publication"},{"issue":"1","key":"1266_CR5","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s10957-015-0704-1","volume":"166","author":"FACC Fontes","year":"2015","unstructured":"Fontes, F.A.C.C., Frankowska, H.: Normality and nondegeneracy for optimal control problems with state constraints. J. Optim. Theory Appl. 166(1), 115\u2013136 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"1266_CR6","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.agwat.2009.09.007","volume":"97","author":"S Galelli","year":"2010","unstructured":"Galelli, S., Gandolfi, C., Soncini-Sessa, R., Agostani, D.: Building a metamodel of an irrigation district distributed-parameter model. Agric. Water Manag. 97(2), 187\u2013200 (2010)","journal-title":"Agric. Water Manag."},{"issue":"2","key":"1266_CR7","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1137\/1037043","volume":"37","author":"RF Hartl","year":"1995","unstructured":"Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181\u2013218 (1995)","journal-title":"SIAM Rev."},{"key":"1266_CR8","doi-asserted-by":"crossref","unstructured":"Kalboussi, N., Roux, S., Boumaza, K., Sinfort, C., Rapaport, A.: About modeling and control strategies for scheduling crop irrigation. IFAC-PapersOnLine, 52(23):43\u201348, (2019). 1st IFAC Workshop on Control Methods for Water Resource Systems CMWRS 2019","DOI":"10.1016\/j.ifacol.2019.11.007"},{"key":"1266_CR9","doi-asserted-by":"crossref","unstructured":"Lopes, S.\u00a0O., Fontes, F.\u00a0A. C.\u00a0C., Caldeira, A.\u00a0C.\u00a0D., and Pereira, R.\u00a0M.\u00a0S.: Optimal control of irrigation with field capacity modes: Characterizing the minimal water consumption solution. In 2018 13th APCA International Conference on Automatic Control and Soft Computing (CONTROLO), pages 300\u2013305, (2018)","DOI":"10.1109\/CONTROLO.2018.8514518"},{"key":"1266_CR10","doi-asserted-by":"crossref","unstructured":"Lopes, S.\u00a0O., Fontes, F.\u00a0A. C.\u00a0C., Pereira, R.\u00a0M.\u00a0S., Pinho, M.\u00a0de., Gon\u00e7alves, A.\u00a0M.: Optimal control applied to an irrigation planning problem. Math. Probl. Eng., 2016:10 pages, (2016)","DOI":"10.1155\/2016\/5076879"},{"issue":"2","key":"1266_CR11","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1504\/IJHST.2019.098161","volume":"9","author":"SO Lopes","year":"2019","unstructured":"Lopes, S.O., Pereira, R.M.S., Pereira, P.A.: Optimal control applied to an irrigation planning problem: a real case study in portugal. Int. J. Hydrol. Sci. Technol. 9(2), 173\u2013188 (2019)","journal-title":"Int. J. Hydrol. Sci. Technol."},{"issue":"3","key":"1266_CR12","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1137\/0315023","volume":"15","author":"H Maurer","year":"1977","unstructured":"Maurer, H.: On optimal control problems with bounded state variables and control appearing linearly. SIAM J. Control. Optim. 15(3), 345\u2013362 (1977)","journal-title":"SIAM J. Control. Optim."},{"issue":"4","key":"1266_CR13","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1093\/imamci\/16.4.335","volume":"16","author":"F Rampazzo","year":"1999","unstructured":"Rampazzo, F., Vinter, R.B.: A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control. IMA J. Math. Control Inform. 16(4), 335\u2013351 (1999)","journal-title":"IMA J. Math. Control Inform."},{"issue":"2","key":"1266_CR14","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1002\/oca.740","volume":"25","author":"U Shani","year":"2004","unstructured":"Shani, U., Tsur, Y., Zemel, A.: Optimal dynamic irrigation schemes. Optimal Control Appl. Methods 25(2), 91\u2013106 (2004)","journal-title":"Optimal Control Appl. Methods"},{"key":"1266_CR15","unstructured":"Vinter, R.: Optimal control. Systems & Control: Foundations & Applications. Birkh\u00e4user Boston, Inc., Boston, MA, (2000)"},{"key":"1266_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.conengprac.2020.104407","volume":"100","author":"N Zeng","year":"2020","unstructured":"Zeng, N., Cen, L., Xie, Y., Zhang, S.: Nonlinear optimal control of cascaded irrigation canals with conservation law pdes. Control. Eng. Pract. 100, 104407 (2020)","journal-title":"Control. Eng. Pract."},{"key":"1266_CR17","unstructured":"Zotarelli, L., Dukes, M.\u00a0D., Romero, C.\u00a0C., Migliaccio, K.\u00a0W., Morgan, K.\u00a0T.: Step by step calculation of the Penman\u2013Monteith evapotranspiration (FAO\u201356 Method). Technical report, UF\/IFAS Extension \u2013 AE459, (2021)"}],"container-title":["International Journal of Applied and Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40819-022-01266-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40819-022-01266-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40819-022-01266-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,13]],"date-time":"2022-04-13T03:52:24Z","timestamp":1649821944000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40819-022-01266-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,11]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,4]]}},"alternative-id":["1266"],"URL":"https:\/\/doi.org\/10.1007\/s40819-022-01266-9","relation":{},"ISSN":["2349-5103","2199-5796"],"issn-type":[{"type":"print","value":"2349-5103"},{"type":"electronic","value":"2199-5796"}],"subject":[],"published":{"date-parts":[[2022,3,11]]},"assertion":[{"value":"2 February 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 March 2022","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"On behalf of all authors, the corresponding author states that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"For numerical computations, a TI-<i>n<\/i>spire CX II-T CAS graphic calculator and MATLAB R2019a tools have been used.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}},{"value":"The authors state that this article complies with ethical standards.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical statement"}}],"article-number":"67"}}