On the componentwise stability of linear systems. (English) Zbl 1056.93054

Summary: The componentwise asymptotic stability (CWAS) and componentwise exponential asymptotic stability (CWEAS) represent stronger types of asymptotic stability, which were first defined for symmetrical bounds constraining the flow of the state-space trajectories, and then, were generalized for arbitrary bounds, not necessarily symmetrical. Our paper explores the links between the symmetrical and the general case, proving that the former contains all the information requested by the characterization of the CWAS/CWEAS as qualitative properties. Complementary to the previous approaches to CWAS/CWEAS that were based on the construction of special operators, we incorporate the flow-invariance condition into the classical framework of stability analysis. Consequently, we show that the componentwise stability can be investigated by using the operator defining the system dynamics, as well as the standard \(\varepsilon\)-\(\delta\) formalism. Although this paper explicitly refers only to continuous-time linear systems, the key elements of our work also apply, mutatis mutandis, to discrete-time linear systems.


93D20 Asymptotic stability in control theory
Full Text: DOI


[1] Free response characterization via flow-invariance. Preprints of the 9th IFAC World Congress, vol. 5, Budapest, 1984; 12-17.
[2] Voicu, IEEE Transactions on Automatic Control 29 pp 937– (1984)
[3] Hmamed, International Journal of Robust and Nonlinear Control 7 pp 1023– (1997)
[4] Hmamed, Automatica 33 pp 1759– (1997)
[5] Matrix Analysis. Cambridge University Press: Cambridge, 1985. · Zbl 0576.15001
[6] Feedback Systems: Input-Output Properties. Academic Press: New York, 1975. · Zbl 0327.93009
[7] Qualitative Theory of Dynamical Systems. Marcel Dekker: New York, 1995.
[8] Kiendl, IEEE Transactions on Automatic Control 37 pp 839– (1992)
[9] Stability, Asymptotic Behavior of Differential Equations. DC Heath: Boston, 1965.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.