{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T10:50:22Z","timestamp":1760784622451},"reference-count":33,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2019,3,13]],"date-time":"2019-03-13T00:00:00Z","timestamp":1552435200000},"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>In this paper, the constrained controllability of the second order retarded nonlinear systems with nonlocal condition has been established by using the theory of cosine families and the generalized open mapping theorem. A new set of sufficient conditions for the constrained controllability of retarded nonlinear systems is established under the assumption that the associated linear system is controllable. By using the Banach fixed point theorem, the existence of mild solution for the considered system with nonlocal delay condition has been deduced. Finally, an example is provided to verify the effectiveness of the obtained theory.<\/jats:p>","DOI":"10.1093\/imamci\/dnz007","type":"journal-article","created":{"date-parts":[[2019,2,22]],"date-time":"2019-02-22T12:37:37Z","timestamp":1550839057000},"page":"441-454","source":"Crossref","is-referenced-by-count":10,"title":["Constrained controllability of second order retarded nonlinear systems with nonlocal condition"],"prefix":"10.1093","volume":"37","author":[{"given":"Suman","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, IGNTU Amarkantak, Anuppur, Madhya Pradesh, India"}]},{"given":"R","family":"Sakthivel","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Bharathiar University, Coimbatore, Tamilnadu, India"}]}],"member":"286","published-online":{"date-parts":[[2019,3,13]]},"reference":[{"key":"2020091709242247100_ref1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.amc.2015.01.118","article-title":"Approximate controllability of second order semilinear stochastic system with nonlocal conditions","volume":"258","author":"Arora","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"2020091709242247100_ref2","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s41980-018-0043-8","article-title":"Exact null controllability of Sobolev-type Hilfer fractional stochastic differential equations with fractional Brownian motion and Poisson jumps","volume":"44","author":"Ahmed","year":"2018","journal-title":"Bull. Iranian Math. Soc."},{"key":"2020091709242247100_ref3","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1186\/s13662-018-1679-7","article-title":"Impulsive Hilfer fractional differential equations","volume":"2018","author":"Ahmed","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"2020091709242247100_ref4","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1006\/jdeq.2000.3885","article-title":"Structured population models, conservation laws and delay equations","volume":"168","author":"Bocharov","year":"2000","journal-title":"J. Differential Equations"},{"key":"2020091709242247100_ref5","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1016\/0022-247X(91)90164-U","article-title":"Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem","volume":"162","author":"Byszewski","year":"1991","journal-title":"J. Math. Anal. Appl."},{"key":"2020091709242247100_ref6","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-4224-6","volume-title":"An Introduction to Infinite-Dimensional Linear Systems Theory","author":"Curtain","year":"1995"},{"key":"2020091709242247100_ref7","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/BF00940064","article-title":"Controllability questions for nonlinear systems in abstract spaces","volume":"68","author":"Chukwu","year":"1991","journal-title":"J. Optim. Theory Appl."},{"key":"2020091709242247100_ref8","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.chaos.2017.03.023","article-title":"Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces","volume":"102","author":"Debbouche","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"2020091709242247100_ref9","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.amc.2014.05.087","article-title":"Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions","volume":"243","author":"Debbouche","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"2020091709242247100_ref10","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-00547-7","volume-title":"Nonlinear Functional Analysis","author":"Deimling","year":"1985"},{"key":"2020091709242247100_ref11","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1016\/j.amc.2009.08.036","article-title":"Existence and regularity of solutions to nonlocal retarded differential equations","volume":"215","author":"Dubey","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"2020091709242247100_ref12","volume-title":"Second Order Linear Differential Equations in Banach Spaces, vol. 108","author":"Fattorini","year":"1985"},{"key":"2020091709242247100_ref13","doi-asserted-by":"crossref","first-page":"3333","DOI":"10.1016\/j.na.2011.02.010","article-title":"Existence of solutions of non-autonomous second order functional differential equations with infinite delay","volume":"74","author":"Henr\u00edquez","year":"2011","journal-title":"Nonlinear Anal."},{"key":"2020091709242247100_ref14","doi-asserted-by":"crossref","first-page":"2372","DOI":"10.1002\/mma.2983","article-title":"Approximate controllability of second order distributed systems","volume":"37","author":"Henr\u00edquez","year":"2014","journal-title":"Math. Methods Appl. Sci."},{"key":"2020091709242247100_ref15","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1016\/j.na.2008.01.029","article-title":"Approximate controllability of second-order distributed implicit functional systems","volume":"70","author":"Henr\u00edquez","year":"2009","journal-title":"Nonlinear Anal."},{"key":"2020091709242247100_ref16","first-page":"189","article-title":"Controllability of linear dynamical systems","volume":"1","author":"Kalman","year":"1963","journal-title":"Contrib. Differential Equations"},{"key":"2020091709242247100_ref17","doi-asserted-by":"crossref","DOI":"10.1016\/j.sysconle.2007.10.002","article-title":"Controllability of second order infinite-dimensional systems","volume":"57","author":"Klamka","year":"2008","journal-title":"Systems Control Lett."},{"key":"2020091709242247100_ref18","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1006\/jmaa.1996.0260","article-title":"Constrained controllability of nonlinear systems","volume":"201","author":"Klamka","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"2020091709242247100_ref19","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s11071-008-9389-4","article-title":"Constrained controllability of semilinear systems with delays","volume":"56","author":"Klamka","year":"2009","journal-title":"Nonlinear Dynam."},{"key":"2020091709242247100_ref20","first-page":"111","article-title":"Constrained controllability of second order dynamical systems with delay","volume":"42","author":"Klamka","year":"2013","journal-title":"Control Cybernet."},{"key":"2020091709242247100_ref21","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10883-017-9376-5","article-title":"Controllability of the second order nonlinear differential equations with non-instantaneous impulses","volume":"24","author":"Kumar","year":"2018","journal-title":"J. Dynam. Control Systems"},{"key":"2020091709242247100_ref22","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1080\/01630563.2013.814067","article-title":"Controllability of second-order systems with nonlocal conditions in Banach spaces","volume":"35","author":"Kumar","year":"2014","journal-title":"Numer. Funct. Anal. Optim."},{"key":"2020091709242247100_ref33","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/s10883-016-9355-2","article-title":"Mild solution and constrained local controllability of semilinear boundary control systems","volume":"23","author":"Kumar","year":"2017","journal-title":"J. Dyn. Control Syst."},{"key":"2020091709242247100_ref23","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/dny037","article-title":"Mild solution and controllability of second-order non-local retarded semilinear systems","author":"Kumar","year":"2018","journal-title":"IMA J. Math. Control Inform."},{"key":"2020091709242247100_ref24","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1023\/A:1023639908792","article-title":"Approximate controllability for a class of abstract second order functional evolution equations","volume":"117","author":"McKibben","year":"2003","journal-title":"J. Optim. Theory Appl."},{"key":"2020091709242247100_ref25","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1080\/00036819708840609","article-title":"Global existence for second order semilinear ordinary and delay integrodifferential equations with nonlocal conditions","volume":"67","author":"Ntouyas","year":"1997","journal-title":"Appl. Anal."},{"key":"2020091709242247100_ref26","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1080\/01630560802580901","article-title":"Approximate controllability of second order systems with state-dependent delay","volume":"29","author":"Sakthivel","year":"2008","journal-title":"Numer. Funct. Anal. Optim."},{"key":"2020091709242247100_ref27","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.na.2008.10.029","article-title":"On controllability of second order nonlinear impulsive differential systems","volume":"71","author":"Sakthivel","year":"2009","journal-title":"Nonlinear Anal."},{"key":"2020091709242247100_ref28","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.sysconle.2017.04.013","article-title":"Constrained controllability of fractional linear systems with delays in control","volume":"106","author":"Sikora","year":"2017","journal-title":"Systems Control Lett."},{"key":"2020091709242247100_ref29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-247X(90)90192-I","article-title":"A unified approach to constrained approximate controllability for the heat equations and the retarded equations","volume":"150","author":"Son","year":"1990","journal-title":"J. Math. Anal. Appl."},{"key":"2020091709242247100_ref30","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s12591-012-0147-1","article-title":"On controllability of nonlocal retarded semilinear distributed control systems","volume":"21","author":"Tomar","year":"2013","journal-title":"Differential Equations Dynam. Systems"},{"key":"2020091709242247100_ref31","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF01902205","article-title":"Cosine families and abstract nonlinear second order differential equations","volume":"32","author":"Travis","year":"1978","journal-title":"Acta Math. Hungar."},{"key":"2020091709242247100_ref32","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.18514\/MMN.2017.2396","article-title":"Null controllability of nonlocal Hilfer fractional stochastic differential equations","volume":"18","author":"Wang","year":"2017","journal-title":"Miskolc Math. Notes"}],"container-title":["IMA Journal of Mathematical Control and Information"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/imamci\/article-pdf\/37\/2\/441\/33762497\/dnz007.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/imamci\/article-pdf\/37\/2\/441\/33762497\/dnz007.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T11:28:09Z","timestamp":1662982089000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/imamci\/article\/37\/2\/441\/5373064"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,13]]},"references-count":33,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,3,13]]},"published-print":{"date-parts":[[2020,6,6]]}},"URL":"https:\/\/doi.org\/10.1093\/imamci\/dnz007","relation":{},"ISSN":["0265-0754","1471-6887"],"issn-type":[{"value":"0265-0754","type":"print"},{"value":"1471-6887","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2020,6]]},"published":{"date-parts":[[2019,3,13]]}}}