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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.

In 1, the authors study the stability of the system
$x¨t+Dx˙t+Ktxt=θn,x0≕x0,x˙0≕x˙0,$
(1)
for all $t⩾0,$ where $xt∈Rn$ and $θn$ is the zero vector in $Rn.$ In (1), the constant damping matrix $D∈Rn×n$ is symmetric and positive definite and so is the stiffness matrix $Kt$ for all $t⩾0.$ Moreover, the function $t↦Kt$ is continuously differentiable and there exist positive constants $k0$ and $k1$ such that
$k0In⩽Kt⩽k1In,$
(2)
for all $t⩾0,$ where $In$ is the $n×n$ identity matrix.

The authors let $σD={d1,d2,…,dn}$ denote the set of eigenvalues (spectrum) of the matrix D where the elements of this set are ordered as $0 They denote the maximum eigenvalue of a symmetric matrix H by $λmaxH.$

With this setup, the stability result in 1 is:

Theorem 1. Let
$α≔maxlimt⩾0{λmaxK−1/2tK˙tK−1/2t},$
(3)

$D^≔D−d1In.$
(4)
If (i)
$D^Kt+KtD^⩾0,$
(5)
for all $t⩾0$ and (ii)
$d1>α2,$
(6)
then the system (1) is exponentially stable.□

It should be noted that for a general function $t↦Kt,$ it is straightforward to derive $t↦K˙t$ in the closed form; however, it is not as such for the function $t↦K−1/2t.$ Therefore, when Theorem 1 is used, α in (3) should be computed numerically by computing $λmaxK−1/2tK˙tK−1/2t$ at instances $t=0,$$t1,$$t2,…,ti,…,$ 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 $D^$ 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:

Theorem 2. Consider the system
$x¨t+Dtx˙t+Ktxt=θn,x0≕x0,x˙0≕x˙0,$
(7)
for all $t⩾0,$ where $xt∈Rn.$ Let the function $t↦Dt$ be piecewise continuous, the matrix $Kt$ be symmetric and positive definite for all $t⩾0,$ and the function $t↦K1/2t$ be differentiable. If
$Pt+PTt⩽0,$
(8)
for all $t⩾0,$ where
$Pt≔−K−1/2t[K1/2ts˙+DtK1/2t],$
(9)
then the system (1) is stable.□
When Theorem 2 is used, the matrix $Pt$ in (9) should be computed numerically at instances $t=0,t1,t2,…,ti,…,$ which are separated from each other by a suitable step size h, by the formula
$Pti=−K−1/2tiK1/2ti+1−K1/2titi+1−ti+DtiK1/2ti.$
(10)
Having computed $Pt$ at $t=0,$$t1,$$t2,…,ti,…,$ the truth of inequality (8) can be examined (numerically).

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.

Consider the system
$x¨1tx¨2t+0.60.050.050.5x˙1tx˙2t+1+2t1+t2001+2t1+t2x1tx2t=θ2,$
(11)
for all $t⩾0.$ Identifying the matrices D and $Ks˙$ in (11), the following can be obtained
$d1=0.4793,d2=0.6207,$
(12)

$K−1/2tK˙tK−1/2t=21+2t1+t0021+2t1+t,$
(13)
for all $t⩾0.$ From (13), it is clear that α=2, and hence inequality (6) does not hold. That is, Theorem 1 cannot determine the stability of the system (11). Roughly speaking, Theorem 1 could have established the stability of the system (11) if the system were more damped, i.e., if $d1$ were larger.
Next, using (9), it is concluded that for the system (11),
$Pt=−11+2t1+t0011+2t1+t−0.60.050.050.5,$
(14)
for all $t⩾0.$ Having $Ps˙$ in (14), it is easily concluded that inequality (8) holds. Thus, by Theorem 2, the system (11) is stable. In fact, it is clear from (14) that for any positive definite matrix D, no matter how small its eigenvalues are, inequality (8) holds, and the stability of the system (11) is guaranteed. Roughly speaking, with a very small damping (dissipation), the system (11) is stable.

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.

1.
Gil’, M. I., Stability of Finite and Infinite Dimensional Systems, Kluwer Academic, Norwell, MA, 1998.