{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:30:33Z","timestamp":1787329833006,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["G0A5317N"],"award-info":[{"award-number":["G0A5317N"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61773140"],"award-info":[{"award-number":["61773140"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004497","name":"Onderzoeksraad, KU Leuven","doi-asserted-by":"publisher","award":["C14\/17\/072"],"award-info":[{"award-number":["C14\/17\/072"]}],"id":[{"id":"10.13039\/501100004497","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential equation. This boundary value problem can be seen as a natural generalization of the classical Lyapunov matrix equation. Lyapunov matrices play an important role in constructing Lyapunov functionals and in $\\mathcal{H}_2$ optimal control. In this paper we present a general approach for computing delay Lyapunov matrices and $\\mathcal{H}_2$ norms for systems with multiple discrete delays, whose applicability extends toward problems where the matrices are large and sparse, and the associated positive semidefinite matrix has a low rank. The problems addressed are challenging, because the boundary value problem is matrix valued with a structure that is much harder to exploit than in the delay-free case, and its solution is in the generic situation nonsmooth. In contrast to existing methods that are based on solving the boundary value problem directly, our method is grounded in solving standard Lyapunov equations of increased dimensions. It combines several ingredients: (i) a spectral discretization of the system of delay equations, (ii) a targeted similarity transformation which induces a desired structure and sparsity pattern and, at the same time, favors accurate low-rank solutions of the corresponding Lyapunov equation, and (iii) a Krylov method for large-scale matrix Lyapunov equations. The structure of the problem is exploited in such a way that the final algorithm does not involve a preliminary discretization step and provides a fully dynamic construction of approximations of increasing rank. Interpretations are also given in terms of a projection method directly applied to a standard linear infinite-dimensional system, which is equivalent to the original time-delay system. Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are reported to illustrate the effectiveness of the algorithms.<\/jats:p>","DOI":"10.1137\/18m1209842","type":"journal-article","created":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T10:58:13Z","timestamp":1562151493000},"page":"845-869","source":"Crossref","is-referenced-by-count":12,"title":["Computing Delay Lyapunov Matrices and $\\mathcal{H}_2$ Norms for Large-scale Problems"],"prefix":"10.1137","volume":"40","author":[{"given":"Wim","family":"Michiels","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bin","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,7,3]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803438523"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/030601600"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-009-0233-7"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/15M1032296"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2015.2487478"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/100813257"},{"key":"atypb7","first-page":"804","volume":"64","author":"Gomez M. A.","year":"2019","journal-title":"IEEE Trans. Automat. Control"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"J. K. Hale and S. M. Verduyn Lunel,\n                      Introduction to Functional Differential Equations\n                      , Appl. Math. Sci. 99, Springer, New York, 1993.","DOI":"10.1007\/978-1-4612-4342-7"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"E. Huesca, S. Mondi\u00e9, and O. Santos,\n                      Polynomial approximations of the Lyapunov matrix of a class of time delay systems\n                      , in Proceedings of the 8th IFAC Workshop on Time-Delay Systems, 2009, pp. 261-266.","DOI":"10.3182\/20090901-3-RO-4009.00042"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s00498-012-0096-9"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/10078270X"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/15M1044667"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2018.08.011"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2010.2067510"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"V. L. Kharitonov,\n                      Time-delay systems. Lyapunov Functionals and Matrices\n                      , Birkh\u00e4user, Basel, 2013.","DOI":"10.1007\/978-0-8176-8367-2"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.sysconle.2006.01.005"},{"key":"atypb17","unstructured":"M. Krsti\u0107,\n                      Delay Compensation for Nonlinear, Adaptive and PDE Systems\n                      , Birkh\u00e4user, Basel, 2007."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1049\/iet-cta.2010.0752"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/100797436"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"W. Michiels and S. Niculescu,\n                      Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue Based Approach\n                      , 2nd ed., SIAM, Philadelphia, 2014.","DOI":"10.1137\/1.9781611973631"},{"key":"atypb21","unstructured":"S.I. Niculescu,\n                      Delay Effects on Stability. A Robust Control Approach\n                      , Lecture Notes in Control and Inform. Sci. 269, Springer, New York, 2001."},{"key":"atypb22","first-page":"1","author":"Peeters J.","year":"2012","journal-title":"France"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/06066120X"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/130912839"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"L. N. Trefethen,\n                      Spectral Methods in MATLAB\n                      , Software Environments Tools 10, SIAM, Philadelphia, 2000.","DOI":"10.1137\/1.9780898719598"},{"key":"atypb26","unstructured":"L. N. Trefethen,\n                      Approximation Theory and Approximation Practice\n                      , SIAM, Philadelphia, 2013."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/140976698"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1080\/00207179.2010.547222"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2008.2008345"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2011.12.009"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1209842","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:35:54Z","timestamp":1787326554000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1209842"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1209842"],"URL":"https:\/\/doi.org\/10.1137\/18m1209842","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}