{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T21:43:27Z","timestamp":1778622207601,"version":"3.51.4"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2014,2,1]],"date-time":"2014-02-01T00:00:00Z","timestamp":1391212800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2014,12]]},"DOI":"10.1007\/s11075-013-9822-8","type":"journal-article","created":{"date-parts":[[2014,1,31]],"date-time":"2014-01-31T22:00:11Z","timestamp":1391205611000},"page":"771-792","source":"Crossref","is-referenced-by-count":24,"title":["Numerical algorithm for a class of constrained optimal control problems of switched systems"],"prefix":"10.1007","volume":"67","author":[{"given":"Xiang","family":"Wu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kanjian","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changyin","family":"Sun","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2014,2,1]]},"reference":[{"key":"9822_CR1","doi-asserted-by":"crossref","first-page":"2526","DOI":"10.1016\/j.automatica.2009.07.018","volume":"45","author":"W Zhang","year":"2009","unstructured":"Zhang, W., Abate, A., Hu, J.H., Vitus, M.: Exponential stabilization of discrete\u2013time switched linear systems. Autom. 45, 2526\u20132536 (2009)","journal-title":"Autom."},{"key":"9822_CR2","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1002\/oca.892","volume":"31","author":"C Kirches","year":"2010","unstructured":"Kirches, C., Sager, S., Bock, H.G., Schloder, J.P.: Time-optimal control of automobile test drives with gear shifts. Optim. Control Appl. Meth. 31, 137\u2013153 (2010)","journal-title":"Optim. Control Appl. Meth."},{"key":"9822_CR3","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1049\/sb:20045019","volume":"1","author":"R Ghosh","year":"2004","unstructured":"Ghosh, R., Tomlin, C.: Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modelling: Delta-Notch protein signalling. Syst. Biol. 1, 170\u2013183 (2004)","journal-title":"Syst. Biol."},{"key":"9822_CR4","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1109\/9.911417","volume":"46","author":"C.G Cassandras","year":"2001","unstructured":"Cassandras, C.G., Pepyne, D.L., Wardi, Y.Y.: Optimal control of a class of hybrid systems. IEEE Trans. Autom. Control. 46, 398\u2013415 (2001)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR5","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.automatica.2003.10.022","volume":"40","author":"H Rehbinder","year":"2004","unstructured":"Rehbinder, H., Sanfridson, M.: Scheduling of a limited communication channel for optimal control. Autom. 40, 491\u2013500 (2004)","journal-title":"Autom."},{"key":"9822_CR6","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/S0005-1098(98)00178-2","volume":"35","author":"A Bemporad","year":"1999","unstructured":"Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Autom. 35, 407\u2013428 (1999)","journal-title":"Autom."},{"key":"9822_CR7","first-page":"11","volume":"41","author":"SC Bengea","year":"2005","unstructured":"Bengea, S.C., DeCarlo, R.A.: Optimal control of switching systems. Autom. 41, 11\u201327 (2005)","journal-title":"Autom."},{"key":"9822_CR8","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1109\/TAC.2003.821417","volume":"49","author":"XP Xu","year":"2004","unstructured":"Xu, X.P., Antsaklis, P.J.: Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans. Autom. Control. 49, 2\u201316 (2004)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR9","first-page":"433","volume":"15","author":"M Boccadoro","year":"2005","unstructured":"Boccadoro, M., Wardi, Y.Y., Egerstedt, M., Verriest, E.: Optimal control of switching surfaces in hybrid dynamical systems. Discrete Event Dyn. Syst.-Theory Appl. 15, 433\u2013448 (2005)","journal-title":"Syst.-Theory Appl."},{"key":"9822_CR10","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/00207170701556880","volume":"81","author":"I Hwang","year":"2008","unstructured":"Hwang, I., Li, J., Du, D.: A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach. Int. J. Control. 81, 277\u2013293 (2008)","journal-title":"Int. J. Control."},{"key":"9822_CR11","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s10957-007-9305-y","volume":"136","author":"H Axelsson","year":"2008","unstructured":"Axelsson, H., Wardi, Y.Y., Egerstedt, M., Verriest, E.: Gradient descent approach to optimal mode scheduling in hybrid dynamical systems. J. Optim. Theory Appl. 136, 167\u2013186 (2008)","journal-title":"J. Optim. Theory Appl."},{"key":"9822_CR12","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1016\/j.automatica.2012.03.019","volume":"48","author":"M Kamgarpour","year":"2012","unstructured":"Kamgarpour, M., Tomlin, C.: On optimal control of non-autonomous switched systems with a fixed mode sequence. Autom. 48, 1177\u20131181 (2012)","journal-title":"Autom."},{"key":"9822_CR13","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1109\/TAC.2010.2085573","volume":"56","author":"D Gorges","year":"2011","unstructured":"Gorges, D., Izak, M., Liu, S.: Optimal control and scheduling of switched systems. IEEE Trans. Autom. Control. 56, 135\u2013140 (2011)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR14","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1080\/00207721.2010.549590","volume":"43","author":"N Cao","year":"2012","unstructured":"Cao, N., Zhang, H.G., Luo, Y.H.: Infinite horizon optimal control of affine nonlinear discrete switched systems using two-stage approximate dynamic programming. Int. J. Syst. Sci. 43, 1673\u20131682 (2012)","journal-title":"Int. J. Syst. Sci."},{"key":"9822_CR15","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1016\/j.sysconle.2012.01.011","volume":"61","author":"M Balandat","year":"2012","unstructured":"Balandat, M., Zhang, W., Abate, A.: On infinite horizon switched LQR problems with state and control constraints. Syst. Control. Lett. 61, 464\u2013471 (2012)","journal-title":"Syst. Control. Lett."},{"key":"9822_CR16","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1016\/j.automatica.2012.03.006","volume":"48","author":"JJ Gao","year":"2012","unstructured":"Gao, J.J., Li, D.: Linear-quadratic switching control with switching cost. Autom. 48, 1138\u20131143 (2012)","journal-title":"Autom."},{"key":"9822_CR17","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1109\/TAC.2008.2010968","volume":"54","author":"W Zhang","year":"2009","unstructured":"Zhang, W., Hu, J.H., Abate, A.: On the value functions of the discrete-time switched LQR problem. IEEE Trans. Autom. Control. 54, 462\u2013471 (2009)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR18","doi-asserted-by":"crossref","first-page":"1815","DOI":"10.1109\/TAC.2011.2178649","volume":"57","author":"W Zhang","year":"2012","unstructured":"Zhang, W., Hu, J.H., Abate, A.: Infinite-horizon switched LQR problems in discrete time: a suboptimal algorithm with performance analysis. IEEE Trans. Autom. Control. 57, 1815\u20131821 (2012)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR19","doi-asserted-by":"crossref","unstructured":"Wu, X., Zhang, K.J., Sun, C.Y.: Constrained optimal control of switched systems and its application. Optim. (2013). doi: 10.1080\/02331934.2013.784764","DOI":"10.1080\/02331934.2013.784764"},{"key":"9822_CR20","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1007\/s10957-013-0306-8","volume":"159","author":"X Wu","year":"2013","unstructured":"Wu, X., Zhang, K.J., Sun, C.Y.: Parameter tuning of multi-proportional-integral-derivative controllers based on optimal switching algorithms. J. Optim. Theory. Appl. 159, 454\u2013472 (2013)","journal-title":"J. Optim. Theory. Appl."},{"key":"9822_CR21","first-page":"61","volume":"22","author":"V Azhmyakov","year":"2012","unstructured":"Azhmyakov, V., Basin, M.V., Raisch, J.: A proximal point based approach to optimal control of affine switched systems. Discrete Event Dyn. Syst.-Theory. Appl. 22, 61\u201381 (2012)","journal-title":"Syst.-Theory. Appl."},{"key":"9822_CR22","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/9.654885","volume":"43","author":"MS Branicky","year":"1998","unstructured":"Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control. 43, 31\u201345 (1998)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR23","first-page":"1453","volume":"62","author":"XP Xu","year":"2005","unstructured":"Xu, X.P., Antsaklis, P.J.: On time optimal control of integrator switched systems with state constraints. Nonlinear Anal.\u2013Theory Meth. Appl. 62, 1453\u20131465 (2005)","journal-title":"Appl."},{"key":"9822_CR24","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.automatica.2010.10.010","volume":"47","author":"TM Caldwell","year":"2011","unstructured":"Caldwell, T.M., Murphey, T.D.: Switching mode generation and optimal estimation with application to skid-steering. Autom. 47, 50\u201364 (2011)","journal-title":"Autom."},{"key":"9822_CR25","doi-asserted-by":"crossref","first-page":"1592","DOI":"10.1109\/TAC.2011.2174694","volume":"57","author":"TM Caldwell","year":"2012","unstructured":"Caldwell, T.M., Murphey, T.D.: Single integration optimization of linear time-varying switched systems. IEEE Trans. Autom. Control. 57, 1592\u20131597 (2012)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR26","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1109\/TAC.2005.861711","volume":"51","author":"M Egerstedt","year":"2006","unstructured":"Egerstedt, M., Wardi, Y.Y., Axelsson, H.: Transition-time optimization for switched-mode dynamical systems. IEEE Trans. Autom. Control. 51, 110\u2013115 (2006)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR27","doi-asserted-by":"crossref","first-page":"2692","DOI":"10.1016\/j.automatica.2009.07.028","volume":"45","author":"P Howlett","year":"2009","unstructured":"Howlett, P., Pudney, P., Vu, X.: Local energy minimization in optimal train control. Autom. 45, 2692\u20132698 (2009)","journal-title":"Autom."},{"key":"9822_CR28","doi-asserted-by":"crossref","first-page":"591","DOI":"10.3934\/jimo.2012.8.591","volume":"8","author":"CH Jiang","year":"2012","unstructured":"Jiang, C.H., Teo, K.L., Loxton, R., Duan, G.R.: A neighboring extremal solution for an optimal switched impulsive control problem. J. Ind. Manage. Optim. 8, 591\u2013609 (2012)","journal-title":"J. Ind. Manage. Optim."},{"key":"9822_CR29","doi-asserted-by":"crossref","first-page":"2455","DOI":"10.1109\/TAC.2009.2029310","volume":"54","author":"R Loxton","year":"2009","unstructured":"Loxton, R., Teo, K.L., Rehbock, V.: Computational method for a class of switched system optimal control problems. IEEE Trans. Autom. Control. 54, 2455\u20132460 (2009)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR30","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1016\/j.automatica.2008.10.031","volume":"45","author":"R Loxton","year":"2009","unstructured":"Loxton, R., Teo, K.L., Rehbock, V., Ling, W.: Optimal switching instants for a switched-capacitor DC\/DC power converter. Autom. 45, 973\u2013980 (2009)","journal-title":"Autom."},{"key":"9822_CR31","doi-asserted-by":"crossref","first-page":"1587","DOI":"10.1109\/TAC.2007.904451","volume":"52","author":"M Shaikh","year":"2007","unstructured":"Shaikh, M., Caines, P.: On the hybrid optimal control problem: Theory and algorithms. IEEE Trans. Autom. Control. 52, 1587\u20131603 (2007)","journal-title":"IEEE Trans. Autom. Control."},{"key":"9822_CR32","volume-title":"Theory and algorithms for indirect methods in optimal control of hybrid systems","author":"B Passenberg","year":"2012","unstructured":"Passenberg, B.: Theory and algorithms for indirect methods in optimal control of hybrid systems. PhD thesis, Technical University of Munich (2012)"},{"key":"9822_CR33","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1002\/oca.880","volume":"30","author":"K Graichen","year":"2009","unstructured":"Graichen, K., Petit, N.: Incorporating a class of constraints into the dynamics of optimal control problems. Optim. Control. Appl. Meth. 30, 537\u2013561 (2009)","journal-title":"Optim. Control. Appl. Meth."},{"key":"9822_CR34","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/0005-1098(82)90086-3","volume":"18","author":"Y Sakawa","year":"1982","unstructured":"Sakawa, Y., Shindo, Y.: Optimal control of container cranes. Autom. 18, 257\u2013266 (1982)","journal-title":"Autom."},{"key":"9822_CR35","doi-asserted-by":"crossref","first-page":"985","DOI":"10.2514\/1.47458","volume":"33","author":"M Soler","year":"2010","unstructured":"Soler, M., Olivares, A., Staffetti, E.: Hybrid optimal control approach to commercial aircraft trajectory planning. J. Guid. Control. Dyn. 33, 985\u2013991 (2010)","journal-title":"J. Guid. Control. Dyn."},{"key":"9822_CR36","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0663-7","volume-title":"Optimization: Algorithms and Consistent Approximations","author":"E Polak","year":"1997","unstructured":"Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, Berlin (1997)"},{"key":"9822_CR37","doi-asserted-by":"crossref","unstructured":"Gonzalez, H., Vasudevan, R., Kamgarpour, M., Sastry, S., Bajcsy, R., Tomlin, C.: A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems. In: Johansson K., Yi W. (eds.) 13th International Conference On Hybrid Systems: Computation And Control, 12\u201316 April 2010. Lecture Notes in Computer Science, pp. 5160. Stockholm (2010)","DOI":"10.1145\/1755952.1755961"},{"key":"9822_CR38","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/BF01581153","volume":"66","author":"R Fletcher","year":"1994","unstructured":"Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program 66, 327\u2013349 (1994)","journal-title":"Math. Program"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-013-9822-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-013-9822-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-013-9822-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T05:48:29Z","timestamp":1565156909000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-013-9822-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,1]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2014,12]]}},"alternative-id":["9822"],"URL":"https:\/\/doi.org\/10.1007\/s11075-013-9822-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,2,1]]}}}