Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-01T04:57:27.877Z Has data issue: false hasContentIssue false
  • English
  • Français

The solution of Lyapunov's matrix equation by a geometric method

Published online by Cambridge University Press:  14 November 2011

N. J. Young
N. J. Young
Affiliation:
Department of Mathematics, University of Glasgow
Get access Rights & Permissions [Opens in a new window]

Synopsis

The author has recently proposed a new algorithm for the solution of the Lyapunov matrix equation of stability theory. This algorithm is based on a formula for the solution of a special case of the equation. This formula is established in the present paper by means of a geometric interpretation. The key ideas are the uses of shift operators and non-orthogonal projections in infinite-dimensional Hilbert space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 3-4 , 1980 , pp. 347 - 354
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Barnett, S. and Storey, C.. Matrix Methods in Stability Theory (London: Nelson, 1970).Google Scholar
2Belanger, P. R. and McGillvray, T. P.. Computational experience with the solution of the matrix Lyapunov equation. IEEE Trans. Automatic Control 21 (1976), 799–800.CrossRefGoogle Scholar
3Bedel'baev, A. Ya.. Ustoichivost' nelineinych sistem avtomaticheskogo regulirovaniya (Stability of non-linear automatic control systems), (Alma-Ata: Izdat. Akad. Nauk Kazah. SSR, 1960).Google Scholar
4Bitmead, R. R. and Weiss, H., On the solution of the discrete time Lyapunov matrix equation in controllable canonical form. IEEE Trans. Automatic Control 24 (1979), 481–482.Google Scholar
5Krein, M. G. and Daletskii, Yu. L., Stability of Solutions of Differential Equations in a Banach Space. Amer. Math. Soc. Math. Monographs 43 (1974).Google Scholar
6Lancaster, P.. Theory of Matrices (New York: Academic Press, 1969).Google Scholar
7Pace, I. S. and Barnett, S.. Comparison of numerical methods for solving Liapunov matrix equations. Internat. J. Control 15 (1972), 907–915.Google Scholar
8Pták, V. and Young, N. J.. Functions of operators and the spectral radius. Linear Algebra and Appl. 29 (1980), 357–392.CrossRefGoogle Scholar
9Smith, R. A.. Matrix calculations for Liapunov quadratic forms. J. Differential Equations 2 (1966), 208–217.Google Scholar
10Sui-Lin, Tsai. The formula of Liapunoff function of system of linear differential equations with constant coefficients. Acta Math. Sinica 9 (1959), 455–467.Google Scholar
11Young, N. J.. Formulae for the solution of Lyapunov matrix equations. Internat. J. Control 31 (1980), 159–179.CrossRefGoogle Scholar
12Young, N. J.. An identity which implies Cohn's theorem on the zeros of a polynomial. J. Math. Anal. Appl 70 (1979), 240–248.CrossRefGoogle Scholar