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Quadratic forms as Liapunov functions for linear differential equations with real constant coefficients

Published online by Cambridge University Press:  24 October 2008

Siegfried H. Lehnigk
Siegfried H. Lehnigk
Affiliation:
Physical Sciences Laboratory, U.S. Army Missile Command, Alabama
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Abstract

This paper deals with the existence of a quadratic form as a Liapunov function for a linear homogeneous vector differential equation with constant coefficient matrix such that the total derivative of the Liapunov function is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. A theorem is proved which guarantees the existence of a strictly semi-definite quadratic form which is not identically equal to 0 for every non-trivial solution if and only if the coefficient matrix of the equation is not proportional to the unit matrix. Furthermore, it is proved that if the equilibrium of the given linear equation is asymptotically stable and if the coefficient matrix is not proportional to the unit matrix, then there exists a positive definite quadratic form as a Liapunov function whose total derivative is strictly negative semi-definite and not identically equal to 0 for every non-trivial solution of the given equation. It is also shown how this theorem fits into the system of already known theorems concerned with quadratic forms as Liapunov functions for a linear equation with constant coefficient matrix.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 4 , October 1965 , pp. 883 - 888
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

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