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| EXISTENCE CONDITIONS OF A SPECIAL TYPE OF LIAPUNOV FUNCTIONAL FOR TWO-DIMENSIONAL DELAY SYSTEM |
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Citation： |
Huang Wenzhang.EXISTENCE CONDITIONS OF A SPECIAL TYPE OF LIAPUNOV FUNCTIONAL FOR TWO-DIMENSIONAL DELAY SYSTEM[J].Chinese Annals of Mathematics B,1984,5(4):711~720 |
| Page view： 705
Net amount： 820 |
Authors： |
Huang Wenzhang; |
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| Abstract： |
In this paper, the author considers the two-dimensional delay systems
$$\[\mathop x\limits^ \cdot (t) = Ax(t) + Bx(t - r),A,B \in {R^{2 \times 2}},x \in {R^2},r = const \ge 0\]$$
and gives the necessary and sifficient conditions under which where exists a simple type of positive definite Liapunov functional
$$\[V(\varphi ) \buildrel \Delta \over = {\varphi ^'}(0){T_\varphi }(0) + \int_{ - \tau }^0 {{\varphi ^'}(\theta )E\varphi (\theta )d\theta } \]$$
and $\[\alpha (s)\]$(where T , E are positive definite 2x2 matrices, $\[\varphi \in C([ - \tau ,0],{R^n})\]$, "." stands for transpose, $\[\alpha (s)\]$ is continuous and $\[\alpha (0) = 0,\alpha (s) > 0,s > 0\]$. such that $\[{V_{(*)}}(\varphi ) \le - \alpha (\left| {\varphi (0)} \right|).\]$. |
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