{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:18:23Z","timestamp":1787321903847,"version":"build-2736575974"},"reference-count":49,"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":[[2015,1]]},"abstract":"<jats:p>We aim at characterizing the asymptotic behavior of value functions in the control of piecewise deterministic Markov processes (PDMP) of switch type under nonexpansive assumptions. For a particular class of processes inspired by temperate viruses, we show that uniform limits of discounted problems as the discount decreases to zero and time-averaged problems as the time horizon increases to infinity exist and coincide. The arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics. The approach strongly relies on viscosity techniques, linear programming arguments, and coupling via random measures associated to PDMP. As an intermediate step in our approach, we present the approximation of discounted value functions when using piecewise constant (in time) open-loop policies.<\/jats:p>","DOI":"10.1137\/140998913","type":"journal-article","created":{"date-parts":[[2015,7,16]],"date-time":"2015-07-16T13:25:18Z","timestamp":1437053118000},"page":"1860-1891","source":"Crossref","is-referenced-by-count":7,"title":["Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses"],"prefix":"10.1137","volume":"53","author":[{"given":"Dan","family":"Goreac","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,7,16]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012999364474"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S0294-1449(97)80134-5"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/S0294-1449(99)80019-5"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1080\/03605309808821413"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s002459911022"},{"key":"atypb6","unstructured":"J. P. Aubin and H. Frankowska,\n                      Set-Valued Analysis\n                      , Birkha\u0308user, Boston, 1990."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2002002"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1080\/07362999708809460"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1214\/ECP.v17-1932"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2005021"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-007-0893-6"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1017\/S0269964805050011"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"P. Bre\u0301maud,\n                      Point Processes and Queues: Martingale Dynamics\n                      , Springer Ser. Statist., Springer-Verlag, New York, 1981.","DOI":"10.1007\/978-1-4684-9477-8"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-013-9230-4"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-004-0805-3"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.95.26.15641"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/6.4.375"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1261670695"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/080718541"},{"key":"atypb20","first-page":"415","author":"Costa O. L. V.","year":"2012","journal-title":"NJ"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"O. L. V. Costa and F. Dufour,\n                      Continuous Average Control of Piecewise Deterministic Markov Processes\n                      , Springer Briefs Math., Springer, New York, 2013.","DOI":"10.1007\/978-1-4614-6983-4"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1214\/11-AAP814"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"A. Crudu, A. Debussche, and O. Radulescu,\n                      Hybrid stochastic simplifications for multiscale gene networks\n                      , BMC Systems Biol. (2009), p. 3:89.","DOI":"10.1186\/1752-0509-3-89"},{"key":"atypb24","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1111\/j.2517-6161.1984.tb01308.x","volume":"46","author":"Davis M. H. A.","year":"1984","journal-title":"J. Roy. Statist. Soc. Ser. B"},{"key":"atypb25","first-page":"140","volume":"78","author":"M. H.","journal-title":"Lecture Notes in Control and Informal Sci."},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"M. H. A. Davis,\n                      Markov Models and Optimization\n                      , Monogr. Statist. Appl. Probab. 49, Chapman & Hall, London, 1993.","DOI":"10.1007\/978-1-4899-4483-2"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01204702"},{"key":"atypb28","first-page":"545","volume":"222","author":"Dufour F.","year":"2008","journal-title":"J. Risk Reliability"},{"key":"atypb29","unstructured":"W. Feller,\n                      An Introduction to Probability Theory and Its Applications Vol.\n                      2, 2nd ed., John Wiley, New York, 1971."},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1023\/B:ACAP.0000031200.76583.75"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1007\/BF01215846"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1051\/cocv\/2010103"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-012-9169-x"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1051\/proc\/201445017"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1214\/09-AAP614"},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"G. H. Hardy and J. E. Littlewood,\n                      Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive\n                      , Proc. London Math. Soc., s2-13 (1914), pp. 174-191.","DOI":"10.1112\/plms\/s2-13.1.174"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.040411297"},{"key":"atypb38","unstructured":"N. Ikeda and S. Watanabe,\n                      Stochastic Differential Equations and Diffusion Processes\n                      , North-Holland Math. Library 24, North-Holland, Amsterdam, 1981."},{"key":"atypb39","unstructured":"M. Jacobsen,\n                      Point Process Theory and Applications. Marked Point and Piecewise Deterministic Processes\n                      , Birkha\u0308user, Basel, 2006."},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v4-39"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/s004400050264"},{"key":"atypb42","unstructured":"P.L. Lions, G. Papanicolaou, and S. R. S. Varadhan,\n                      Homogenization of Hamilton-Jacobi Equations\n                      , manuscript, 1987."},{"key":"atypb43","first-page":"199","author":"Oliu-Barton M.","year":"2013","journal-title":"Boston"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1137\/090756818"},{"key":"atypb45","doi-asserted-by":"crossref","unstructured":"J. Renault,\n                      Uniform value in dynamic programming\n                      , J. Euro. Math. Soc. (JEMS), 13 (2011), pp. 309-330.","DOI":"10.4171\/jems\/254"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2009.03.005"},{"key":"atypb47","unstructured":"T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels,\n                      Stochastic Processes for Insurance and Finance\n                      , Wiley Ser. Probab. Statist. 505, John Wiley, New York, and Sons, 2009."},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1137\/0324067"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1007\/s00422-010-0384-8"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/140998913","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:16:04Z","timestamp":1787318164000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/140998913"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":49,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/140998913"],"URL":"https:\/\/doi.org\/10.1137\/140998913","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}