Formal study of the stability of systems started more than a century ago by the pioneering work of Routh, Poincare´, and Lyapunov. Since then many researchers have been vigorously developing the theory of the stability of systems. As a result, presently there is a vast body of literature on this subject. Thus, it may be impossible for an individual researcher to know a good portion of the available results. This can possibly lead to the establishment or rediscovery of stability results whose stronger versions have been already obtained. Such is more or less the case for the stability result in 1.
The authors let denote the set of eigenvalues (spectrum) of the matrix D where the elements of this set are ordered as They denote the maximum eigenvalue of a symmetric matrix H by
With this setup, the stability result in 1 is:
It should be noted that for a general function it is straightforward to derive in the closed form; however, it is not as such for the function Therefore, when Theorem 1 is used, α in (3) should be computed numerically by computing at instances which are separated from each other by a suitable step size h. Moreover, when Theorem 1 is used, the eigenvalues of the damping matrix D should be computed, the matrix be formed, and the truth of inequality (5) be verified (numerically).
A stability result for the system (1), even when the damping matrix is time varying is given in Gil’ (1998, p. 70). This result is:
It appears that in terms of the computational efficiency the stability test in Theorem 1 is not superior to that in Theorem 2. Thus, what should be examined is the applicability of Theorems 1 and 2 in establishing the stability of systems. In the following, these theorems are applied to an example.
In conclusion, the stability result in Gil’ (1998, p. 70) is stronger than that in 1, because (i) it can handle time varying damping matrices; (ii) it can establish the stability of systems whose stability cannot be determined by the result in 1, as it was shown by an example.
By J.-W. Wu and R.-F. Fung and published in the October 1999 issue of the ASME JOURNAL OF VIBRATION AND ACOUSTICS, Vol. 121, No. 4, pp. 509–511.
Research Scientist, Berkeley Engineering Research Institute, P.O. Box 9984, Berkeley, California 94709.