{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T09:26:03Z","timestamp":1790846763459,"version":"4.1.0"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2022,6]]},"abstract":"<jats:p>Discrete algebraic Riccati equations and their fixed points are well understood and arise in a variety of applications; however, the time-varying equations have not yet been fully explored in the literature. In this article we provide a self-contained study of discrete time Riccati matrix difference equations. In particular, we provide a novel Riccati semigroup duality formula and a new Floquet-type representation for these equations. Due to the aperiodicity of the underlying flow of the solution matrix, conventional Floquet theory does not apply in this setting and thus further analysis is required. We illustrate the impact of these formulae with an explicit description of the solution of time-varying Riccati difference equations and its fundamental-type solution in terms of the fixed point of the equation and an invertible linear matrix map as well as uniform upper and lower bounds on the Riccati maps. These are the first results of this type for time-varying Riccati matrix difference equations.<\/jats:p>","DOI":"10.1137\/21m1437226","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T09:34:40Z","timestamp":1653384880000},"page":"1393-1409","source":"Crossref","is-referenced-by-count":6,"title":["A Note on Riccati Matrix Difference Equations"],"prefix":"10.1137","volume":"60","author":[{"given":"Pierre","family":"Del Moral","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emma","family":"Horton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"key":"atypb1","unstructured":"H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank,\n                      Matrix Riccati Equations in Control and Systems Theory\n                      , Birkh\u00e4user, Basel, Switzerland, 2012."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729698"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/16M1102707"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1080\/00207179.2019.1590647"},{"key":"atypb5","doi-asserted-by":"crossref","first-page":"884","DOI":"10.1214\/19-AIHP987","volume":"56","author":"Bishop A. N.","year":"2020","journal-title":"Ann. Inst. Henri Poincar\u00e9 Probab. Statist."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(85)90027-1"},{"key":"atypb7","first-page":"81","author":"Bittanti S.","year":"2009","journal-title":"London"},{"key":"atypb8","unstructured":"The Riccati Equation\n                      , S. Bittanti, A. J. Laub, and J. C. Willems, eds., Springer Science & Business Media, Springer, Berlin, Heidelberg, 2012."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0503065"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2006.10.023"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2011.07.023"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(92)90012-Y"},{"key":"atypb13","unstructured":"P. Del Moral and E. Horton,\n                      A Theoretical Analysis of One-Dimensional Discrete Generation Ensemble Kalman Particle Filters\n                      , preprint, arXiv:2107.01855 [math.PR], 2021."},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2018.05.005"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2011.03.046"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1968.1099024"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2004.11.008"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.24033\/asens.220"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/1023004"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177730946"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/1031049"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1972.1099902"},{"key":"atypb23","unstructured":"A. H. Jazwinski,\n                      Stochastic Processes and Filtering Theory\n                      , Academic Press, Cambridge, MA, 1970."},{"key":"atypb24","unstructured":"T. Kailath, A. H. Sayed, and B. Hassibi,\n                      Linear Estimation\n                      , Prentice Hall, Hoboken, NJ, 2000."},{"key":"atypb25","first-page":"102","volume":"5","author":"Kalman R. E.","year":"1960","journal-title":"Bol. Soc. Mat. Mex."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1972.1099861"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1080\/00207177708922266"},{"key":"atypb28","first-page":"430","volume":"8","author":"Kucera V.","year":"1972","journal-title":"Kybernetika"},{"key":"atypb29","first-page":"42","volume":"9","author":"Kucera V.","year":"1973","journal-title":"Kybernetika"},{"key":"atypb30","doi-asserted-by":"crossref","unstructured":"P. Lancaster and L. Rodman,\n                      Algebraic Riccati equations\n                      , Oxford University Press, Oxford, UK, 1995.","DOI":"10.1093\/oso\/9780198537953.001.0001"},{"key":"atypb31","unstructured":"P. S. Maybeck,\n                      Stochastic Models, Estimation and Control: Volume 1\n                      . Academic Press, Cambridge, MA, 1979."},{"key":"atypb32","unstructured":"K. M. Mikkola,\n                      Riccati Equations and Optimal Control of Well-Posed Linear Systems\n                      , preprint, arXiv:1602.08618v1 [math.02], 2016."},{"key":"atypb33","unstructured":"A. Prach,\n                      Faux Riccati Equation Techniques for Feedback Control of Nonlinear and Time-Varying Systems\n                      , Ph.D. thesis, Middle East Technical University, 2015."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1109\/MCS.2014.2385252"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(60)50046-1"},{"key":"atypb36","doi-asserted-by":"crossref","unstructured":"M. Rhudy, Y. Gu, and M. Napolitano,\n                      Relaxation of initial error and noise bounds for stability of GPS\/INS attitude estimation\n                      , in AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, 2012.","DOI":"10.2514\/6.2012-5031"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729893"},{"key":"atypb38","first-page":"194","author":"Sreedhar J.","year":"1993","journal-title":"Proceedings of the Conference on Information Sciences and Systems"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/BF01219774"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1080\/00207179608921876"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1109\/9.489203"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(92)90118-C"},{"key":"atypb43","unstructured":"M. A. 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