{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:21Z","timestamp":1787336901987,"version":"build-2736575974"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2000,1]]},"abstract":"<jats:p>Solutions of the optimal control and $H_\\infty$-control problems for nonlinear affine systems can be found by solving Hamilton--Jacobi equations. However, these first-order nonlinear partial differential equations can, in general, not be solved analytically. This paper introduces an iterative algorithm which solves these equations numerically for points near the origin. The procedure converges to the stabilizing solution exponentially with respect to the iteration variable. The algorithm is implemented on both illustrative and comparative examples.<\/jats:p>","DOI":"10.1137\/s1064827598344315","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"312-329","source":"Crossref","is-referenced-by-count":22,"title":["An Iterative Algorithm for Solving Hamilton--Jacobi Type Equations"],"prefix":"10.1137","volume":"22","author":[{"given":"Jerry","family":"Markman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I. Norman","family":"Katz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-9262-1"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"Alberto Isidori, Attenuation of disturbances in nonlinear control systems, Progr. Systems Control Theory, Vol. 12, Birkh\u00e4user Boston, Boston, MA, 1992, 275\u201330093f:93047","DOI":"10.1007\/978-1-4757-2204-8_20"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8566-9"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/00207176908905867"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0005-1098(71)90008-2"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1109\/TSMC.1979.4310171"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"R. Beard, G. Saridis, and J. Wen,\n                      Improving the performance of stabilizing controls for nonlinear systems\n                      , IEEE Control Systems Magazine, (1996), pp. 27\u201335.","DOI":"10.1109\/37.537206"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"K. A. Wise and J. L. Sedwick,\n                      Missile autopilot design using nonlinear h\u221e\u2010optimal control\n                      , Proceedings of 13th IFAC Symposium on Automatic Control in Aerospace, Palo Alto, CA, 1994.","DOI":"10.1016\/S1474-6670(17)45789-2"},{"key":"R9","unstructured":"J. Markman and I. N. Katz,\n                      Convergence of an iterative algorithm for solving Hamilton Jacobi type equations\n                      , Math. Comp, to appear."},{"key":"R10","unstructured":"J. Markman,\n                      Numerical solutions of the Hamilton\u2010Jacobi equations arising in nonlinear H\u221e and optimalcontrol\n                      , D.Sc. thesis, Washington University, Department of Systems Science and Mathematics, Pullman, WA, 1998."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827598344315","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:57Z","timestamp":1787333757000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827598344315"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,1]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2000,1]]}},"alternative-id":["10.1137\/S1064827598344315"],"URL":"https:\/\/doi.org\/10.1137\/s1064827598344315","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,1]]}}}