Decay estimates of Green's matrices for discrete-time linear periodic systems (Q6618720)

Various mathematical models for periodic phenomena using linear periodic systems are studied in the literature, see for example [\textit{S. Bittanti} and \textit{P. Colaneri}, Periodic systems. Filtering and control. London: Springer (2009; Zbl 1163.93003)], and modern algorithmic and computational issues related to linear periodic systems are presented in [\textit{A. Varga}, Int. J. Control 86, No. 7, 1227--1239 (2013; Zbl 1278.93132)]. In [\textit{G. V. Demidenko} and \textit{A. A. Bondar}, Sib. Math. J. 57, No. 6, 969--980 (2016; Zbl 1370.39002); translation from Sib. Mat. Zh. 57, No. 6, 1240--1254 (2016)], the authors studied the special case of discrete-time linear periodic systems \(B_px_p-A_px_{p-1}=f_p\), where \(B_p=I\) and \(A_p\) are nonsingular matrices. Based on the periodic Schur decomposition in [\textit{A. Bojanczyk} et al., ``Periodic Schur decomposition: algorithms and applications'', in: Proceedings of SPIE 1770, Advanced Signal Processing Algorithms, Architectures, and Implementations III 1992. 31--42 (1992)] of matrices with reordering and solution of generalized periodic Sylvester systems, which allow to block diagonalize periodic matrix sequences according to the spectrum parts of the monodromy matrix inside and outside the unit disc, the authors of this paper study the general case of discrete-time linear periodic systems, where \(A_p\) and \(B_p\) can be singular. Then they give a decay estimates in terms of solutions to the non-projected periodic Lyapunov matrix equations, see [\textit{T. Penzl}, Adv. Comput. Math. 8, No. 1--2, 33--48 (1998; Zbl 0909.65026)], whose decay is tighter\Nthan that given in [\textit{G. V. Demidenko} and \textit{A. A. Bondar}, Sib. Math. J. 57, No. 6, 969--980 (2016; Zbl 1370.39002); translation from Sib. Mat. Zh. 57, No. 6, 1240--1254 (2016)].