{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T19:42:25Z","timestamp":1770752545026,"version":"3.50.0"},"reference-count":40,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2019,4,1]],"date-time":"2019-04-01T00:00:00Z","timestamp":1554076800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,6,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper proposes an efficient adaptive control parameterization method for solving optimal control problems. In this method, mesh density functions are used to generate mesh points. In the first step, the problem is solved by control parameterization on uniform mesh points. Then at each step, the approximate control obtained from the previous step is applied to construct a mesh density function, and consequently a new adapted set of mesh points. Several numerical examples are included to demonstrate that the adaptive control parameterization method is more accurate than a uniform control parameterization one.<\/jats:p>","DOI":"10.1093\/imamci\/dnz010","type":"journal-article","created":{"date-parts":[[2019,3,9]],"date-time":"2019-03-09T20:10:25Z","timestamp":1552162225000},"page":"497-512","source":"Crossref","is-referenced-by-count":6,"title":["Adaptive control parameterization method by density functions for optimal control problems"],"prefix":"10.1093","volume":"37","author":[{"given":"Nastaran","family":"Ejlali","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Tarbiat Modares University, Tehran, Iran"}]},{"given":"Seyed Mohammad","family":"Hosseini","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Tarbiat Modares University, Tehran, Iran"}]}],"member":"286","published-online":{"date-parts":[[2019,4,1]]},"reference":[{"key":"2020091709241824700_ref18","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/0045-7825(86)90084-8","article-title":"Basic principles of feedback and adaptive approaches in the finite element method","volume":"55","author":"Babu\u0161ka","year":"1986","journal-title":"Comput. 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Phys."},{"key":"2020091709241824700_ref1","first-page":"193","article-title":"Survey of numerical methods for trajectory optimization","volume":"21","author":"Betts","year":"1998","journal-title":"Math. Comp."},{"key":"2020091709241824700_ref21","first-page":"3802","article-title":"Initialization of direct transcription optimal control software. Proceedings of the 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475)","author":"Betts","year":"2003"},{"key":"2020091709241824700_ref24","volume-title":"Applied Optimal Control.","author":"Ho","year":"1975"},{"key":"2020091709241824700_ref35","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0098-1354(89)89006-4","article-title":"Simultaneous optimization and solution methods for batch reactor control profiles","volume":"13","author":"Cuthrell","year":"1989","journal-title":"Comput. Chem. Eng."},{"key":"2020091709241824700_ref37","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1016\/0098-1354(94)00086-4","article-title":"Dynamic optimization of constrained chemical engineering problems using dynamic programming","volume":"19","author":"Dadebo","year":"1995","journal-title":"Comput. Chem. Eng."},{"key":"2020091709241824700_ref3","article-title":"Legendre and pseudospectral Legendre approaches for solving optimal control problems. Ph.D. Thesis","volume-title":"Mississippi State University, USA","author":"Elnager","year":"1993"},{"key":"2020091709241824700_ref5","doi-asserted-by":"crossref","first-page":"1843","DOI":"10.1016\/j.automatica.2010.06.048","article-title":"A unified framework for the numerical solution of optimal control problems using pseudospectral methods","volume":"46","author":"Garg","year":"2010","journal-title":"Automatica J. IFAC"},{"key":"2020091709241824700_ref6","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/0005-1098(88)90003-9","article-title":"Control parameterization: a unified approach to optimal control problems with general constraints","volume":"24","author":"Goh","year":"1988","journal-title":"Automatica J. IFAC"},{"key":"2020091709241824700_ref4","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s10589-007-9102-4","article-title":"Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control","volume":"41","author":"Gong","year":"2008","journal-title":"Comput. Optim. Appl."},{"key":"2020091709241824700_ref33","article-title":"Dotcvp: dynamic optimization toolbox with control vector parameterization approach for handling continuous and mixed-integer dynamic optimization problems","author":"Hirmajer","year":"2008"},{"key":"2020091709241824700_ref25","volume-title":"Adaptive Moving Mesh Methods","author":"Huang","year":"2010"},{"key":"2020091709241824700_ref19","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0045-7825(93)90223-K","article-title":"The theory of density representation of finite element meshes. Examples of density operators with quadrilateral elements in the mapped domain","volume":"109","author":"Hugger","year":"1993","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"2020091709241824700_ref26","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1023\/A:1023600422807","article-title":"Computational method for time-optimal switching control","volume":"117","author":"Kaya","year":"2003","journal-title":"J. Optim. Theory Appl."},{"key":"2020091709241824700_ref8","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1007\/s10957-011-9904-5","article-title":"An exact penalty function method for continuous inequality constrained optimal control problem","volume":"152","author":"Li","year":"2011","journal-title":"J. Optim. Theory Appl."},{"key":"2020091709241824700_ref15","doi-asserted-by":"crossref","first-page":"976","DOI":"10.1002\/asjc.1175","article-title":"A control parameterization approach with variable time nodes for optimal control problems","volume":"18","author":"Li","year":"2016","journal-title":"Asian J. Control"},{"key":"2020091709241824700_ref31","doi-asserted-by":"crossref","first-page":"2863","DOI":"10.1109\/TAC.2014.2317293","article-title":"A novel penalty approach for nonlinear dynamic optimization problems with inequality path constraints","volume":"59","author":"Liu","year":"2014","journal-title":"IEEE Trans. Autom. 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J."},{"key":"2020091709241824700_ref10","doi-asserted-by":"crossref","first-page":"275","DOI":"10.3934\/jimo.2014.10.275","article-title":"The control parameterization method for nonlinear optimal control: a survey","volume":"10","author":"Lin","year":"2014","journal-title":"J. Ind. Manag. Optim."},{"key":"2020091709241824700_ref27","doi-asserted-by":"crossref","first-page":"2923","DOI":"10.1016\/j.automatica.2008.04.011","article-title":"Optimal control problems with multiple characteristic time points in the objective and constraints","volume":"44","author":"Loxton","year":"2008","journal-title":"Automatica J. IFAC"},{"key":"2020091709241824700_ref9","doi-asserted-by":"crossref","first-page":"2250","DOI":"10.1016\/j.automatica.2009.05.029","article-title":"Optimal control problems with a continuous inequality constraint on the state and the control","volume":"45","author":"Loxton","year":"2009","journal-title":"Automatica J. IFAC"},{"key":"2020091709241824700_ref28","first-page":"6566","article-title":"Robust suboptimal control of nonlinear systems","volume":"217","author":"Loxton","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"2020091709241824700_ref36","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1002\/bit.260410513","article-title":"Optimization of fed-batch fermentors by iterative dynamic programming","volume":"41","author":"Luus","year":"1993","journal-title":"Biotechnol. Bioeng."},{"key":"2020091709241824700_ref32","doi-asserted-by":"crossref","first-page":"1550","DOI":"10.1002\/aic.690340917","article-title":"Optimal production of secreted protein in fed\u2013batch reactors","volume":"34","author":"Park","year":"1988","journal-title":"AIChE J."},{"key":"2020091709241824700_ref2","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1137\/1015071","article-title":"An historical survey of computational methods in optimal control","author":"Polak","year":"1973","journal-title":"SIAM Rev."},{"key":"2020091709241824700_ref13","volume-title":"Optimization: Algorithms and Consistent Approximations","author":"Polak","year":"2012"},{"key":"2020091709241824700_ref30","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0005-1098(82)90086-3","article-title":"Optimal control of container cranes","volume":"18","author":"Sakawa","year":"1982","journal-title":"Automatica J. IFAC"},{"key":"2020091709241824700_ref12","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1016\/j.compchemeng.2005.02.036","article-title":"Dynamic optimization using adaptive control vector parameterization","volume":"29","author":"Schlegel","year":"2005","journal-title":"Comput. Chem. Eng."},{"key":"2020091709241824700_ref14","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.jprocont.2005.06.008","article-title":"Detection and exploitation of the control switching structure in the solution of dynamic optimization problems","volume":"16","author":"Schlegel","year":"2006","journal-title":"J. Process Control"},{"key":"2020091709241824700_ref17","volume-title":"Theory and implementation of numerical methods based on Runge\u2013Kutta integration for solving optimal control problems. Doctoral Dissertation","author":"Schwartz","year":"1996"},{"key":"2020091709241824700_ref7","first-page":"314","article-title":"The control parameterization enhancing transform for constrained optimal control problems","volume":"40","author":"Teo","year":"1999","journal-title":"ANZIAM J."},{"key":"2020091709241824700_ref40","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s00449-016-1685-7","article-title":"A new sensitivity-based adaptive control vector parameterization approach for dynamic optimization of bioprocesses","volume":"40","author":"Wang","year":"2017","journal-title":"Bioprocess. Biosyst. Eng."},{"key":"2020091709241824700_ref16","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1080\/0305215X.2016.1270276","article-title":"An effective simultaneous approach with variable time nodes for dynamic optimization problems","volume":"49","author":"Xiao","year":"2017","journal-title":"Eng. Optim."},{"key":"2020091709241824700_ref22","article-title":"A density-function based mesh refinement algorithm for solving optimal control problems. Infotech and Aerospace Conference","author":"Zhao","year":"2009"},{"key":"2020091709241824700_ref23","doi-asserted-by":"crossref","first-page":"271","DOI":"10.2514\/1.45852","article-title":"Density functions for mesh refinement in numerical optimal control","volume":"34","author":"Zhao","year":"2011","journal-title":"J. Guid. 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