{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:40:53Z","timestamp":1787323253919,"version":"build-2736575974"},"reference-count":11,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>\n                    In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C\n                    <jats:sup>0<\/jats:sup>\n                    topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.\n                  <\/jats:p>","DOI":"10.1137\/s036301290138866x","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"991-1014","source":"Crossref","is-referenced-by-count":94,"title":["Strong Optimality for a Bang-Bang Trajectory"],"prefix":"10.1137","volume":"41","author":[{"given":"Andrei A.","family":"Agrachev","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianna","family":"Stefani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"PierLuigi","family":"Zezza","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","unstructured":"A. Agrach\u00ebv, R. Gamkrelidze, Symplectic geometry for optimal control, Monogr. Textbooks Pure Appl. Math., Vol. 133, Dekker, New York, 1990, 263\u201327791f:49032"},{"key":"R2","unstructured":"A. Agrachev, R. Gamkrelidze, Symplectic methods for optimization and control, Monogr. Textbooks Pure Appl. Math., Vol. 207, Dekker, New York, 1998, 19\u2013771493010"},{"key":"R3","unstructured":"V. I. Arnold,\n                      Mathematical Methods in Classical Mechanics\n                      , Springer, New York, 1980."},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/002071798221533"},{"key":"R5","first-page":"8","volume":"220","author":"Agrachev A.","year":"1998","journal-title":"Tr. Mat. Inst. Steklova"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"Andrei Agrachev, Gianna Stefani, Pierluigi Zezza, A Hamiltonian approach to strong minima in optimal control, Proc. Sympos. Pure Math., Vol. 64, Amer. Math. Soc., Providence, RI, 1999, 11\u20132299i:49020","DOI":"10.1090\/pspum\/064\/1654537"},{"key":"R7","unstructured":"M. Giaquinta and S. Hildebrandt,\n                      Calculus of Variations I\n                      , Springer, Berlin, 1996."},{"key":"R8","unstructured":"M. Giaquinta and S. Hildebrandt,\n                      Calculus of Variations II\n                      , Springer, Berlin, 1996."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012999322031"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(92)90076-5"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012993246191"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S036301290138866X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:51:08Z","timestamp":1787320268000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S036301290138866X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":11,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S036301290138866X"],"URL":"https:\/\/doi.org\/10.1137\/s036301290138866x","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}