Abstract
This paper considers the strong stabilization problem: given a linear time-varying system which is stabilizable by dynamic feedback, when can the stabilizer be chosen to be itself stable? We consider here the case of algebras of discrete time, time-varying systems which are asymptotically time-invariant, in the sense that as time evolves the time-varying transfer operator converges to a time-invariant transfer operator. Convergence here is in the sense of uniform or strong convergence of sequences of operators on an appropriate Hilbert space of input–output signals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arveson W (2002) A short course on spectral theory. Graduate texts in mathematics, vol 209. Springer, New York
Barria J, Halmos PR (1982) Asymptotic Toeplitz operators. Trans Am Math Soc 273: 621–630
Bass H (1967) Algebraic K-theory. Benjamin, New York
Cioranescu I, Lizama C (1997) Some applications of Fejer’s Theorem to operator cosine functions in Banach space. Proc Am Math Soc 125(8): 2353–2362
Corach G, Larotonda AR (1984) Stable range in Banach algebras. J Pure Appl Algebra 32: 289–300
Davidson KR, Levene RH, Marcoux LW, Radjavi H (2008) On the topological stable rank of non-selfadjoint operator algebras. Math Ann 34: 239–253
FeintuchA Saeks R (1982) System theory, a Hilbert space approach. Pure and applied mathematics, vol 102. Academic Press, New York
Feintuch A (1998) Robust control theory in Hilbert space. Applied mathematical sciences, vol 130. Springer, Berlin
Feintuch A (2005) On strong stabilization for linear time-varying systems. Syst Control Lett 54: 1091–1095
Feintuch A (1989) On asymptotic Toeplitz and Hankel operators. Oper Theory Adv Appl 41: 241–254
Feintuch A (2009) The stable rank of a nest algebra and strong stabilization of linear time-varying systems. Oper Theory Adv Appl 197: 139–148
Quadrat A (2004) On a general structure of the stabilizing controllers based on a stable range. SIAM J Cont Optim 24(6): 2264–2285
Rabinovich V, Roch S, Silbermann B (2004) Limit Operators and Their Applications in Operator. Theory Operator theory: advances and applications, vol 150. Birkhauser Verlag, Basel
Sasane A (2009) Stable ranks of Banach algebras of operator valued analytic functions. Complex Anal Oper Theory 3: 323–330
Sasane A (2009) Algebras of holomorphic functions and control theory. Dover Publications, New York
Takahashi K (1955) Invertible completions of operator matrices. Integ Eq Oper Theory 21: 355–361
Treil S (1992) The stable rank of the algebra H ∞ equals 1. J Funct Anal 109: 130–154
Vasershtein LN (1972) Stable ranks of rings and dimensionality of topological spaces. Func Anal Appl 5: 102–110
Vidyasagar M (1985) Control System synthesis: a factorization approach. MIT Press, Cambridge,MA
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feintuch, A. On strong stabilization of asymptotically time-invariant linear time-varying systems. Math. Control Signals Syst. 22, 229–243 (2011). https://doi.org/10.1007/s00498-011-0057-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-011-0057-8