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A Remark on a Theorem of Lyapunov

Published online by Cambridge University Press:  20 November 2018

James S. W. Wong
James S. W. Wong*
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania
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Consider the linear ordinary differential equation

1

where x ∊ En, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:

Theorem (Lyapunov). The following three statements are equivalent:

(I) The spectrum σ(A) of A lies in the negative half plane.

(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies

2

where ∥ ∥ denotes the Euclidean norm.

(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there exist q1,q2>0 such that

3

satisfying

4

where I is the identity matrix.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 141 - 143
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Almkvist, G., Stability of linear differential equations in Banach algebras, Math. Scand. 14 (1964), 39-44.Google Scholar
2. Bellman, R., Notes on matrix theory X, A problem in control, Quart. Appl. Math. 14 (1957), 417-419.Google Scholar
3. Datko, R., An extension of a theorem of A. M. Lyapunov to semi-groups of operators, J. Math. Anal. Appl. 24 (1968), 290-295.Google Scholar
4. Gantmacher, F. R., The theory of matrices, English translation, (2 volumes), Chelsea, New York, 1959.Google Scholar
5. Massera, J. L. and Schaffer, J. J., Linear differential equations and functional analysis, III. Lyapunov"s second method in the case of conditional stability, Ann. of Math. 69 (1959), 535-574.Google Scholar
6. Stone, M. H., Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Coll. XV, New York, (1932).Google Scholar