On The Ψ – Asymptotic Stability Of Nonlinear Lyapunov Matrix Differential Equations

Aurel Diamandescu

doi:10.2478/v10324-012-0001-8

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On The Ψ – Asymptotic Stability Of Nonlinear Lyapunov Matrix Differential Equations

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 50 (2012): Issue 1 (November 2012)

By: Aurel Diamandescu

Open Access

|Nov 2012

Abstract

In this paper are proved (necessary and) sufficient conditions for Ψ - asymptotic stability of the trivial solution of onlinear Lyapunov matrix differential equations.

References

[1] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill Book Company, Inc. New York (translated in Romanian), 1960.
Search in Google Scholar
[2] L. Cesari, Un nuovo criterio di stabilita per le soluzioni delle equazioni differenziali lineari, Ann. Scuola Norm. Sup. Pisa, 2, 9, (1940), 163 - 186.
Search in Google Scholar
[3] W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.
Search in Google Scholar
[4] A. Diamandescu, On the Ψ Asymptotic Stability of a Nonlinear Volterra Integro-Differential System, Bull. Math. Soc. Sc. Math., Tome 46(94) No. 1 - 2, (2003), 39 - 60.
Search in Google Scholar
[5] A. Diamandescu, Y - bounded solutions for a Lyapunov matrix differential equation, Electronic Journal of Qualitative Theory of Differential Equations, No. 17, 1 -11. URL: http://www.math.u-szeged.hu/ejqtde/, (2009.)10.14232/ejqtde.2009.1.17
Search in Google Scholar
[6] A. Diamandescu, On Ψ stability of nonlinear Lyapunov matrix differential equations, Electronic Journal of Qualitative Theory of Differential Equations, No. 54, 1 - 18. URL: http://www.math.u-szeged.hu/ejqtde/, (2009)10.14232/ejqtde.2009.1.54
Search in Google Scholar
[7] J. L. Massera and J. J. Sch¨affer, Linear differential equations and functional analysis, I, Ann. of math. 67, (1958), 517 - 573.10.2307/1969871
Search in Google Scholar
[8] M. S. N. Murty and B. V. Apparao, On two point boundary value problems for ˙X = AX + XB, Ultra Science, 16(2)M, (2004), 223 - 227.
Search in Google Scholar
[9] M. S. N. Murty and B. V. Apparao, Two point boundary value problems for matrix differential equations, Journal of the Indian Math. Soc., Vol. 73, Nos. 1-2, (2006), 1-7.10.1155/BVP/2006/83910
Search in Google Scholar
[10] M. S. N. Murty and B. V. Apparao, Kronecker product boundary value problems - existence and uniqueness, Journal of Indian Acad. Mathematics, 25, No. 2, (2003), 383 - 394.
Search in Google Scholar
[11] M. S. N. Murty, B. V. Apparao, and G. Suresh Kumar, Controllability, observability and realizability of matrix Lyapunov systems, Bull. Korean Math. Soc., 43, No. 1, (2006), 149 - 159.10.4134/BKMS.2006.43.1.149
Search in Google Scholar
[12] K. N. Murty, K. R. Prasad, and P. V. S. Anand, Two point boundary value problem associated with Lyapunov type matrix difference system, Dyn. Syst. and Appl., 4, (1995), 205 - 214.
Search in Google Scholar
[13] M. S. N. Murty and G. Suresh Kumar, On Ψboundedness and stability of matrix Lyapunov systems, J. Appl. Math. Comput, 26: 67 - 84, (2008.)10.1007/s12190-007-0007-2
Search in Google Scholar
[14] M. S. N. Murty and G. Suresh Kumar, On dichotomy and conditioning for two-point boundary value problems associated with first order matrix Lyapunov systems, J. Korean Math. Soc., 45, No. 5, (2008), 1361 - 1378.10.4134/JKMS.2008.45.5.1361
Search in Google Scholar
[15] O. Perron, Die Stabilit¨atsfrage bei Differentialgleichungen, Math. Z., 32, 703 - 728., (1930.)10.1007/BF01194662
Search in Google Scholar