# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Matrix measures in the qualitative analysis of parametric uncertain systems. (English) Zbl 1179.93154
Summary: The paper considers parametric uncertain systems of the form $\dot x(t)= Mx(t)$, $M\in{\cal M}$, ${\cal M}\subset\Bbb R^{n\times n}$, where ${\cal M}$ is either a convex hull, or a positive cone of matrices, generated by the set of vertices ${\cal V}=\{M_1,M_2,\dots,M_K\}\subset\Bbb R^{n\times n}$. Denote by $\mu_{\|\ \|}$ the matrix measure corresponding to a vector norm $\|\ \|$. When ${\cal M}$ is a convex hull, the condition $\mu_{\|\ \|}(M_k)\le r<0$, $1\le k\le K$, is necessary and sufficient for the existence of common strong Lyapunov functions and exponentially contractive invariant sets with respect to the trajectories of the uncertain system. When ${\cal M}$ is a positive cone, the condition $\mu_{\|\ \|}(M_k)\le 0$, $1\le k\le K$, is necessary and sufficient for the existence of common weak Lyapunov functions and constant invariant sets with respect to the trajectories of the uncertain system. Both Lyapunov functions and invariant sets are described in terms of the vector norm $\|\ \|$ used for defining the matrix measure $\mu_{\|\ \|}$. Numerical examples illustrate the applicability of our results.

##### MSC:
 93D30 Scalar and vector Lyapunov functions 93C15 Control systems governed by ODE 93C05 Linear control systems
Full Text:
##### References:
 [1] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Company, Boston, Mass, USA, 1965. · Zbl 0154.09301 [2] T. Ström, “Minimization of norms and logarithmic norms by diagonal similarities,” Computing, vol. 10, no. 1-2, pp. 1-7, 1972. · Zbl 0251.65034 · doi:10.1007/BF02242378 [3] C. Desoer and H. Haneda, “The measure of a matrix as a tool to analyze computer algorithms for circuit analysis,” IEEE Transactions on Circuits Theory, vol. 19, no. 5, pp. 480-486, 1972. [4] H. Kiendl, J. Adamy, and P. Stelzner, “Vector norms as Lyapunov functions for linear systems,” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 839-842, 1992. · Zbl 0760.93070 · doi:10.1109/9.256362 [5] Y. Fang and T. G. Kincaid, “Stability analysis of dynamical neural networks,” IEEE Transactions on Neural Networks, vol. 7, no. 4, pp. 996-1006, 1996. [6] L. Gruyitch, J.-P. Richard, P. Borne, and J.-C. Gentina, Stability Domains, vol. 1 of Nonlinear Systems in Aviation, Aerospace, Aeronautics, and Astronautics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. [7] Z. Zahreddine, “Matrix measure and application to stability of matrices and interval dynamical systems,” International Journal of Mathematics and Mathematical Sciences, no. 2, pp. 75-85, 2003. · Zbl 1016.15017 · doi:10.1155/S0161171203202295 · eudml:50145 [8] O. Pastravanu and M. Voicu, “On the componentwise stability of linear systems,” International Journal of Robust and Nonlinear Control, vol. 15, no. 1, pp. 15-23, 2005. · Zbl 1056.93054 · doi:10.1002/rnc.964 [9] G. Söderlind, “The logarithmic norm. History and modern theory,” BIT Numerical Mathematics, vol. 46, no. 3, pp. 631-652, 2006. · Zbl 1102.65088 · doi:10.1007/s10543-006-0069-9 [10] J. Chen, “Sufficient conditions on stability of interval matrices: connections and new results,” IEEE Transactions on Automatic Control, vol. 37, no. 4, pp. 541-544, 1992. · Zbl 0752.15015 · doi:10.1109/9.126595 [11] M. E. Sezer and D. D. \vSiljak, “On stability of interval matrices,” IEEE Transactions on Automatic Control, vol. 39, no. 2, pp. 368-371, 1994. · Zbl 0800.93981 · doi:10.1109/9.272336 [12] O. Pastravanu and M. Voicu, “Necessary and sufficient conditions for componentwise stability of interval matrix systems,” IEEE Transactions on Automatic Control, vol. 49, no. 6, pp. 1016-1021, 2004. · doi:10.1109/TAC.2004.829639 [13] T. Alamo, R. Tempo, D. R. Ramírez, and E. F. Camacho, “A new vertex result for robustness problems with interval matrix uncertainty,” Systems & Control Letters, vol. 57, no. 6, pp. 474-481, 2008. · Zbl 1154.93023 · doi:10.1016/j.sysconle.2007.11.003 [14] A. Michel, K. Wang, and B. Hu, Qualitative Theory of Dynamical Systems. The Role of Stability Preserving Mappings, vol. 239 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2001. · Zbl 0976.34001 [15] X. Liao, L. Q. Wang, and P. Yu, Stability of Dynamical Systems, Elsevier, Amsterdam, The Netherlands, 2007. [16] F. Blanchini and S. Miani, Set-Theoretic Methods in Control, Systems & Control: Foundations & Applications, Birkhäuser, Boston, Mass, USA, 2008. · Zbl 1140.93001 [17] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0704.15002 [18] Y. Mori, T. Mori, and Y. Kuroe, “On a class of linear constant systems which have a common quadratic Lyapunov function,” in Proceedings of the 37th IEEE Conference on Decision & Control, vol. 3, pp. 2808-2809, Tampa, Fla, USA, 1998. [19] L. Kolev and S. Petrakieva, “Assessing the stability of linear time-invariant continuous interval dynamic systems,” IEEE Transactions on Automatic Control, vol. 50, no. 3, pp. 393-397, 2005. · doi:10.1109/TAC.2005.843857 [20] M. H. Matcovschi and O. Pastravanu, “Perron-Frobenius theorem and invariant sets in linear systems dynamics,” in Proceedings of the 15th IEEE Mediterranean Conference on Control and Automation (MED ’07), Athens, Greece, 2007.