Abstract
In this paper the concepts of componentwise asymptotic stability with respect to a differentiable vector function h(t) (approaching 0 as t → ∞) (CWASh) and componentwise exponentially asymptotic stability (CWEAS), previously introduced, have been extended to Q-CWASh and Q-CWEAS (Q being a q × n real matrix), respectively, in order to cover the more general situation of polyhedral time-dependent flow-invariant sets, defined by |Qx| ≤ h(t), x ∈ ℝn, t ∈ ℝ+, symmetrical with respect to the equilibrium point of a given continuous-time linear system ẋ= Ax, t ∈ ℝ+, x ∈ ℝn. It is proved that Q-CWASh is equivalent with the existence of a q × q matrix E such that EQ = QA, Ēh(t) ≤ ḣ(t), where the bar operator (−) transforms only the extra diagonal elements of E into their corresponding absolute values and does not change its diagonal elements. By specializing vector function h(t) in an exponentially decaying form, the concept of Q-CWEAS is characterized by the above mentioned matrix equation and an algebraic inequality. For Q = I n these results consistently yield the earlier ones. As in this case, there exists a strong connection between Q-CWASh (Q-CWEAS) and the asymptotic stability, but now this connection is amended by the observability of the pair (Q, A).
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35690-7_44
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References
Blanchini, F. (1999), Set invariance in control; survey paper. Automatica, 35, 1747–1767.
Brezis, H. (1970), On a characterization of flow-invariant sets. Comm. Pure Appl. Math., 23, 261–263.
Crandall, M.G. (1972), A generalization of Peano’s existence theorem and flow invariance. Proc. Amer. Math. Soc., 36, 151–155.
Martin, R.H. jr. (1973), Differential equations on closed subsets of a Banach space. Trans. Amer. Math. Soc., 179, 399–414.
Nagumo, M. (1942), Uber die Lage der Integralkurven gewöhnlicher Differentialgleichungen. Proc. Phys. Math. Soc. Japan, 24, 551–559.
Pavel, H.N. (1984), Differential Equations Flow Invariance and Applications. Pitman, Boston, London, Melbourne.
Pâstrâvanu, O., Voicu, M. (1999), Flow-invariant rectangular sets and componentwise asymptotic stability of interval matrix systems. 5th European Control Conference, Karlsruhe, August 31 - Sept. 3, 1999; Proceedings on CD, rubiconAgentur far digitale Medien, Aachen (Germany), 16 p.
P5strâvanu, O., Voicu, M. (2000), Robustness analysis of componentwise asymptotic stability. 16th IMACS World Congress, Lausanne, August 21–25, 2000; Proceedings on CD, ©imacs (ISBN 3 9522075 1 9), 16 p.
Pâstrâvanu, O., Voicu, M. (2000), Preserving componentwise asymptotic stability under disturbances. Revue Roumaine des Sciences Techniques (Académie Roumaine), ser. electr. energ., t. 45, 3, 2000, pp. 413–425.
P5strâvanu, O., Voicu, M. (2001), Dynamics of a class of nonlinear systems under flow-invariance constraints. 9th IEEE Mediterranean Conf. on Control and Automation (MED’01), Dubrovnik, June 27–29, 2001; Proceedings on CD-ROM (ISBN 953 6037 35 1), Book of abstracts (ISBN 953 6037 34 3), IEEE, ©KoREMA (Zagreb), 6 p.
Péstrâvanu, O., Voicu, M. (2001), Componentwise asymptotic stability of a class of nonlinear systems. 1st IFAC Symposium on System Structure and Control (SSSC 01), August 29–31, 2001, Czech Technical University of Prague; Book of Abstracts, p. 28; Proceedings, IFAC, CD-ROM, 078, 6 p.
P5str.vanu, O., Voicu, M. (2001), Flow-invariance in exploring stability for a class of nonlinear uncertain systems. 6th European Control Conference, Porto, September 4–7, 2001, Proceedings, CD-ROM, 6 p.
Péstrâvanu, O., Voicu, M. (2001), Robustness of componentwise asymptotic stability for a class of nonlinear systems. Proceedings of the Romanian Academy, ser. A, vol 1, 1–2, 2001, 61–67.
Pâstrâvanu, O., Voicu, M. (2002), Componentwise asymptotic stability of interval matrix systems. Journal of Differential and Integral Equation, vol. 15, No. 11, 1377–1394.
Voicu, M. (1984), Free response characterization via flow invariance. 9th World Congress of IFAC, Budapest, Preprints, 5, 12–17.
Voicu, M. (1984), Componentwise asymptotic stability of linear constant dynamical systems. IEEE Trans. on Aut. Control, 10, 937–939.
Voicu, M. (1987), On the application of the flow-invariance method in control theory and design. 10th World Congress of IFAC, Munich, Preprints, 8, 364–369.
Vorotnikov, I.V., (2002), Partial stability, stabilization and control: a some recent results.15th World Congress of IFAC, Barcelona, Preprints on CD-ROM, 12 p.
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Voicu, M., Pastravanu, O. (2003). Componentwise Asymptotic Stability Induced by Symmetrical Polyhedral Time-Dependent Constraints. In: Barbu, V., Lasiecka, I., Tiba, D., Varsan, C. (eds) Analysis and Optimization of Differential Systems. SEC 2002. IFIP — The International Federation for Information Processing, vol 121. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35690-7_43
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DOI: https://doi.org/10.1007/978-0-387-35690-7_43
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