{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:42:14Z","timestamp":1787319734792,"version":"build-2736575974"},"reference-count":60,"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":[[2009,1]]},"abstract":"<jats:p>We investigate relationships between the deterministic infinite time horizon optimal control problem with discounting, in which the state trajectories remain in a given compact set Y, and a certain infinite dimensional linear programming (IDLP) problem. We introduce the problem dual with respect to this IDLP problem and obtain some duality results. We construct necessary and sufficient optimality conditions for the optimal control problem under consideration, and we give a characterization of the viability kernel of Y. We also indicate how one can use finite dimensional approximations of the IDLP problem and its dual for construction of near optimal feedback controls. The construction is illustrated with a numerical example.<\/jats:p>","DOI":"10.1137\/070696209","type":"journal-article","created":{"date-parts":[[2009,8,17]],"date-time":"2009-08-17T11:23:29Z","timestamp":1250508209000},"page":"2480-2512","source":"Crossref","is-referenced-by-count":67,"title":["Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting"],"prefix":"10.1137","volume":"48","author":[{"given":"Vladimir","family":"Gaitsgory","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marc","family":"Quincampoix","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,8,14]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"O. Alvarez and M. Bardi,\n                      Ergodic problems in differential games\n                      , in Advances in Dynamic Games Theory, S. Jorgensen, M. Quincampoix, and T.L. Vincent, eds., Birkh\u00e4user Boston, Boston, MA, 2007, pp. 131\u2013152.","DOI":"10.1007\/978-0-8176-4553-3_7"},{"key":"R2","unstructured":"E.J. Anderson and P. Nash,\n                      Linear Programming in Infinite-Dimensional Spaces\n                      , Wiley, Chichester, 1987."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"M. Arisawa, H. Ishii, and P.L. Lions,\n                      A characterization of the existence of solutions for Hamilton\u2013Jacobi equations in ergodic control problems with applications\n                      , Appl. Math. Optim., 42 (2000), pp 35\u201350.","DOI":"10.1007\/s002450010001"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1142\/S0219493704001152"},{"key":"R5","unstructured":"J.P. Aubin,\n                      Viability Theory\n                      , Birkh\u00e4user Boston, Boston, MA, 1991."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2007.12.022"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"M. Bardi and I. Capuzzo-Dolcetta,\n                      Optimal Control and Viscosity Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations\n                      , Systems Control Found. Appl., Birkh\u00e4user Boston, Boston, MA, 1997.","DOI":"10.1007\/978-0-8176-4755-1"},{"key":"R8","unstructured":"A. Bensoussan,\n                      Perturbation Methods in Optimal Control\n                      , John Wiley, New York, 1989."},{"key":"R9","unstructured":"D.P. Bertsekas and S.E. Shreve,\n                      Stochastic Optimal Control: The Discrete Time Case\n                      , Academic Press, New York, 1978."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1080\/07362999708809460"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1065725192"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-007-0893-6"},{"key":"R13","unstructured":"H. Br\u00e9zis,\n                      Analyse fonctionnelle. Th\u00e9orie et applications\n                      , Collection Math\u00e9matiques Appliqu\u00e9es pour la Ma\u00eetrise, Masson, Paris, 1983."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/s00182-006-0030-9"},{"key":"R15","unstructured":"D.A. Carlson, A.B. Haurie, and A. Leizarowitz,\n                      Optimal Control on Infinite Time Horizon\n                      , 2nd ed., Springer-Verlag, Berlin, 1991."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02254655"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"F. Colonius and W. Kliemann,\n                      The Dynamics of Control\n                      , Birkh\u00e4user Boston, Boston, MA, 2000.","DOI":"10.1007\/978-1-4612-1350-5"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00266-5"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-06-04103-1"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012902416080"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2002030"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/060676398"},{"key":"R23","unstructured":"W.H. Fleming and H.M. Soner,\n                      Controlled Markov Processes and Viscosity Solutions\n                      , Springer-Verlag, New York, 1991."},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/0327060"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(89)90203-5"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2000.3711"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0330065"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012903424186"},{"key":"R29","first-page":"329","volume":"13","author":"Gaitsgory V.","year":"2006","journal-title":"J. Convex Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0944-6532","issn-type":"print"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/040616802"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012902417620"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1998.3451"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012997315919"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1023\/A:1004659107996"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.4064\/am-24-1-17-33"},{"key":"R36","doi-asserted-by":"crossref","unstructured":"O. Hernandez-Lerma and J.B. Lasserre,\n                      Markov Chains and Invariant Probabilities\n                      , Birkh\u00e4user-Verlag, Basel, 2003.","DOI":"10.1007\/978-3-0348-8024-4"},{"key":"R37","first-page":"343","volume":"3","author":"Kamenskii M.","year":"2002","journal-title":"Int. J. Pure Appl. Math."},{"key":"R38","unstructured":"E. Klein and A.C. Thompson,\n                      Theory of Correspondence\n                      , Wiley-Interscience, New York, 1984."},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012995295516"},{"key":"R40","doi-asserted-by":"crossref","unstructured":"H.J. Kushner,\n                      Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems\n                      , Birkh\u00e4user Boston, Boston, MA, 1990.","DOI":"10.1007\/978-1-4612-4482-0"},{"key":"R41","doi-asserted-by":"crossref","unstructured":"H.J. Kushner and P.G. Dupuis,\n                      Numerical Methods for Stochastic Control Problems in Continuous Time\n                      , 2nd ed., Springer-Verlag, New York, 2001.","DOI":"10.1007\/978-1-4613-0007-6"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1137\/070685051"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(90)90294-P"},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1023\/A:1024077306226"},{"key":"R45","unstructured":"J.G. Llavona,\n                      Approximation of Continuously Differentiable Functions\n                      , Math. Stud. 130, North\u2013Holland, Amsterdam, The Netherlands, 1986."},{"key":"R46","doi-asserted-by":"crossref","unstructured":"K.R. Parthasarathy,\n                      Probability Measures on Metric Spaces\n                      , Academic Press, New York, 1967.","DOI":"10.1016\/B978-1-4832-0022-4.50006-5"},{"key":"R47","doi-asserted-by":"crossref","unstructured":"S. Pickenhain and V. Lykina,\n                      Sufficiency conditions for infinite horizon optimal control problems\n                      , in Recent Advances in Optimization, Lecture Notes in Econom. and Math. Systems 563, Springer, Berlin, 2006, pp. 217\u2013232.","DOI":"10.1007\/3-540-28258-0_14"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1137\/0330020"},{"key":"R49","doi-asserted-by":"publisher","DOI":"10.1016\/j.bulsci.2004.07.001"},{"key":"R50","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(03)00094-4"},{"key":"R51","unstructured":"J.E. Rubio,\n                      Control and Optimization. The Linear Treatment of Nonlinear Problems\n                      , Manchester University Press, Manchester, UK, 1985."},{"key":"R52","unstructured":"W. Rudin,\n                      Real and Complex Analysis\n                      , 3rd ed., McGraw-Hill, New York, 1987."},{"key":"R53","doi-asserted-by":"crossref","unstructured":"T.R. Rockafellar and R. Wets,\n                      Variational Analysis\n                      , Grundlehren Math. Wiss. 317, Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-642-02431-3"},{"key":"R54","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176990944"},{"key":"R55","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176990945"},{"key":"R56","unstructured":"A. Vigodner,\n                      Limits of Singularly Perturbed Control Problems: Dynamical Systems Approach\n                      , Thesis for the Degree of Doctor of Philosophy, The Weizmann Institute of Science, Rehovot, Israel, 1995."},{"key":"R57","doi-asserted-by":"crossref","unstructured":"C. Villani,\n                      Topics in Optimal Transportation\n                      , Grad. Stud. Math. 58, AMS, Providence, RI, 2003.","DOI":"10.1090\/gsm\/058"},{"key":"R58","doi-asserted-by":"publisher","DOI":"10.1137\/0331024"},{"key":"R59","doi-asserted-by":"crossref","unstructured":"J. Warga,\n                      Optimal Control of Differential and Functional Equations\n                      , Academic Press, New York, 1972.","DOI":"10.1016\/B978-0-12-735150-6.50011-7"},{"key":"R60","unstructured":"G.G. Yin and Q. Zhang,\n                      Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach\n                      , Springer, New York, 1997."}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/070696209","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:51:58Z","timestamp":1787316718000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/070696209"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":60,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/070696209"],"URL":"https:\/\/doi.org\/10.1137\/070696209","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}