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| A Problem on the Stability of Retarded Systems |
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Citation： |
Huang Zhenxun,Lin Xiaobiao.A Problem on the Stability of Retarded Systems[J].Chinese Annals of Mathematics B,1982,3(1):115~120 |
| Page view： 812
Net amount： 785 |
Authors： |
Huang Zhenxun; Lin Xiaobiao |
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| Abstract： |
In this paper, the following retarded system has been studied
$\[\dot x(t) = Ax(t) + Bx(t - r),r > 0\]$(1)
where x(t) is an n-vector valued function; A and B are n*n constant matrices, and all the eigenvalues of A are supposed to have negative real parts. The asymptotical stability of equation (1) has been discussed by Halec13 utilizing the following Liapunov functional
$\[V(\phi ) = {\phi ^T}(0)C\phi (0) + \int_{ - r}^0 {{\phi ^T}(\theta )E\varphi (\theta )} d\theta \]$,
where E>0 and the symmetric matrix C>0 is chosen, such that A^TC+CA= — D<0. In this discussion, he remarked that if matrix
$\[H = \left[ {\begin{array}{*{20}{c}}
{D - E}&{ - CB}\{ - {{(CB)}^T}}&E
\end{array}} \right] > 0\]$,
the rate of decay of the solution of equation (1) to zero would be independent of the delay r, that is, would follow the exponential relation as indicated below : $\[||x(t,{t_0},\phi )|| \le K(r){e^{ - \alpha (t - {t_0})}}||\phi ||\]$,where \alpha(\alpha >0) is indepndent of r.
We show that this conclusion is not true, and a new relation between Liapunov functional and it's solution (exponential estimation) has been developed for the general rOtarded functional differential equation
$\[|\dot X(t) = f(t,{X_t})\]$(2)
If there is a functional $\[V(t,\phi ):{R^ + } \times {C_H} \to R\]$ such that
(i)$\[v|\phi (0){|^\eta } \le V(t,\phi ) \le K||\phi ||_\eta ^\eta ,(v,K > 0,\eta > 0)\]$
(ii)$\[\dot V(t,\phi ) \le - {C_1}|\phi (0){|^\eta },({C_1} > 0)\]$
then the solution of equation (2) x(t_0, ф) (t) satisfies
$\[||x({t_0},\phi )(t)|| \le {K_1}(r)||\phi |{|_\eta }{e^{ - {\alpha _1}(r)(t - {t_0})}}\]$
where \alpha _1 depends on r.
The following inverse problem has also been studied: In case the solution x = 0 of equation (1) is asymptotically stable for every value of r> 0, would there exist the matrices C>0 and E>0 such that the corresponding matrix H>0? Counter example is given for this problem. |
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