{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:33Z","timestamp":1760060133865,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of Hunan Province","award":["2023JJ50164","2023JJ50165","2026JJ50180","JGYB23009","Y2025452"],"award-info":[{"award-number":["2023JJ50164","2023JJ50165","2026JJ50180","JGYB23009","Y2025452"]}]},{"name":"Hunan University of Technology","award":["2023JJ50164","2023JJ50165","2026JJ50180","JGYB23009","Y2025452"],"award-info":[{"award-number":["2023JJ50164","2023JJ50165","2026JJ50180","JGYB23009","Y2025452"]}]},{"name":"Hunan Province","award":["2023JJ50164","2023JJ50165","2026JJ50180","JGYB23009","Y2025452"],"award-info":[{"award-number":["2023JJ50164","2023JJ50165","2026JJ50180","JGYB23009","Y2025452"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Consider a class of coupled discrete-time Riccati equations arising from jump systems. To compute their solutions when systems reach a steady state, we propose an operator Newton method and correspondingly establish its quadratic convergence under suitable assumptions. The advantage of the proposed method lies in the fact that its subproblems are solved using the operator Smith method, which allows it to maintain quadratic convergence in both the inner and outer iterations. Moreover, it does not require the constant term matrix of the equation to be invertible, making it more broadly applicable than existing inverse-free iterative methods. For large-scale problems, we develop a low-rank variant by incorporating truncation and compression techniques into the operator Newton framework. A complexity analysis is also provided to assess its scalability. Numerical experiments demonstrate that the presented low-rank operator Newton method is highly effective in approximating solutions to large-scale structured coupled Riccati equations.<\/jats:p>","DOI":"10.3390\/axioms14080601","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T08:46:55Z","timestamp":1754383615000},"page":"601","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Operator Newton Method for Large-Scale Coupled Riccati Equations Arising from Jump Systems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4742-4364","authenticated-orcid":false,"given":"Bo","family":"Yu","sequence":"first","affiliation":[{"name":"School of Science, Hunan University of Technology, Zhuzhou 412007, China"}]},{"given":"Yiwen","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Science, Hunan University of Technology, Zhuzhou 412007, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1515-8225","authenticated-orcid":false,"given":"Ning","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Science, Hunan University of Technology, Zhuzhou 412007, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1016\/0005-1098(94)00164-E","article-title":"On the solution of discretetime markovian jump linear quadratic control problems","volume":"31","author":"Freiling","year":"1995","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Liu, Y., Wang, Z., and Lin, X. (2023). Non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems with applications. Axioms, 12.","DOI":"10.3390\/axioms12090882"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Do Costa, O.L.V., Marques, R.P., and Fragoso, M.D. (2005). Discrete-Time Markov Jump Linear Systems, Springer.","DOI":"10.1007\/b138575"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.sysconle.2008.12.004","article-title":"Gradient based iterative algorithm for solving coupled matrix equations","volume":"58","author":"Zhou","year":"2009","journal-title":"Syst. Control. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1016\/j.camwa.2007.04.031","article-title":"Iterative solutions of coupled discrete Markovian jump Lyapunov equations","volume":"55","author":"Wang","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1109\/TAC.2014.2326273","article-title":"New iterative algorithms for solving coupled Markovian jump Lyapunov equations","volume":"60","author":"Wu","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Dragan, V., Morozan, T., and Stoica, A.M. (2010). Robust Control of Discrete-time Linear Stochastic Systems, Springer.","DOI":"10.1007\/978-1-4419-0630-4"},{"key":"ref_8","first-page":"228","article-title":"Accelerated LMI solvers for the maximal solution to a set of discrete-time algebraic Riccati equations","volume":"12","author":"Ivanov","year":"2012","journal-title":"Appl. Math. E-Notes"},{"key":"ref_9","first-page":"34","article-title":"A method to solve the discrete-time coupled algebraic riccati equations","volume":"206","author":"Ivanov","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1177\/0142331214528350","article-title":"An improved Schulz-type iterative method for matrix inversion with application","volume":"36","author":"Haghani","year":"2014","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"341","DOI":"10.4208\/jcm.1010-m3258","article-title":"On eigenvalue bounds and iteration methods for discrete algebraic riccati equations","volume":"29","author":"Dai","year":"2011","journal-title":"J. Comput. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1109\/TAC.2024.3476993","article-title":"An inversion-free iterative algorithm with a scalar tuning parameter for coupled Riccati matrix equations arising in LQ optimal control of Markov jump systems","volume":"70","author":"Jiang","year":"2024","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4754","DOI":"10.1109\/TAC.2022.3172266","article-title":"An inversion-free iterative algorithm for riccati matrix equations in discrete-time markov jump systems","volume":"67","author":"Li","year":"2022","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1007\/s12190-024-01990-z","article-title":"Newton\u2019s method for coupled continuous-time algebraic Riccati equations","volume":"70","author":"Feng","year":"2024","journal-title":"J. Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1109\/T-C.1974.223788","article-title":"A computational algorithm for solving a system of coupled algebraic matrix riccati equations","volume":"C-23","author":"Salama","year":"1974","journal-title":"IEEE Trans. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Margenov, S.D., Vulkov, L.G., and Wasniewski, J. (2009). Numerical Analysis and Its Applications. Proceedings of the 4th International Conference (NAA 2008), Lozenetz, Bulgaria, 16\u201320 June 2008, Springer. Revised Selected Papers.","DOI":"10.1007\/978-3-642-00464-3"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1093\/imamci\/dnad033","article-title":"A new inversion-free iterative algorithm for the discrete algebraic Riccati equation","volume":"41","author":"Wang","year":"2024","journal-title":"IMA J. Math. Control Inf."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jerbi, H., Alshammari, O., Aoun, S.B., Kchaou, M., Simos, T.E., Mourtas, S.D., and Katsikis, V.N. (2024). Hermitian solutions of the quaternion algebraic Riccati equations through zeroing neural networks with application to quadrotor control. Mathematics, 12.","DOI":"10.3390\/math12010015"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Escorcia, J.M., and Suazo, E. (2024). On blow-up and explicit soliton solutions for coupled variable coefficient nonlinear schr\u00f6dinger equations. Mathematics, 12.","DOI":"10.3390\/math12172694"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Yu, B., Dong, N., and Hu, B.-Q. (2024). Operator Smith algorithm for coupled Stein equations from jump control systems. Axioms, 13.","DOI":"10.3390\/axioms13040249"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1564","DOI":"10.1137\/17M1157155","article-title":"Solving rank structured Sylvester and Lyapunov equations","volume":"39","author":"Massei","year":"2018","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"C43","DOI":"10.1137\/19M1288048","article-title":"hm-toolbox: Matlab software for HODLR and HSS matrices","volume":"42","author":"Massei","year":"2020","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Higham, N.J. (2002). Accuracy and Stability of Numerical Algorithms, SIAM. [2nd ed.].","DOI":"10.1137\/1.9780898718027"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"111057","DOI":"10.1016\/j.automatica.2023.111057","article-title":"Factorized squared Smith method for large-scale Stein equations with high-rank terms","volume":"154","author":"Yu","year":"2023","journal-title":"Automatica"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.cam.2019.04.014","article-title":"Large-scale algebraic Riccati equations with high-rank constant terms","volume":"361","author":"Yu","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Higham, N.J. (2008). Functions of Matrices: Theory and Computation, SIAM.","DOI":"10.1137\/1.9780898717778"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Benner, P., Sorensen, D.C., and Mehrmann, V. (2005). Oberwolfach Benchmark Collection. Dimension Reduction of Large-Scale Systems, Springer. Lecture Notes in Computational Science and Engineering.","DOI":"10.1007\/3-540-27909-1"},{"key":"ref_28","unstructured":"Lang, N. (2018). Numerical Methods for Large-Scale Linear Time-Varying Control Systems and related Differential Matrix Equations, Logos-Verlag."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S1474-6670(17)55030-2","article-title":"Asymptotically Stable All-Pass Transfer Functions: Canonical Form, Parametrization and Realization","volume":"20","author":"Ober","year":"1987","journal-title":"IFAC Proc. Vol."},{"key":"ref_30","unstructured":"Schmidt, A., and Siebert, K. (2005). Design of Adaptive Finite Element Software\u2014The Finite Element Toolbox ALBERTA, Springer. Lecture Notes in Computational Science and Engineering."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/3-540-27909-1_24","article-title":"Benchmark examples for model reduction of linear time-invariant dynamical systems, dimension reduction of large-scale systems","volume":"Volume 45","author":"Chahlaoui","year":"2005","journal-title":"Dimension Reduction of Large-Scale Systems"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/601\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:21:36Z","timestamp":1760034096000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/601"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,1]]},"references-count":31,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080601"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080601","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,8,1]]}}}