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Stability of impulsive control systems with time delay. (English) Zbl 1081.93021
The following time delay impulsive control system is considered $x(t)=Ax(t)+Bx(t-r)$, $t\ne\tau_k$, $x(t^+)=x(t^-)+C_kx(t^-)$, $t= \tau_k$, where $A,B,C_k$, $k=1,2$, are some constant matrices, $r>0$ is a delay constant, $\{\tau_k,C_k(x(\tau_k^-))$, $k=1,2,\dots\}$ denotes the impulsive control law with $0<\tau_1<\tau_2<\cdots<\tau_k <\tau_{k+1}<\dots$, $\tau_k\to\infty$, as $k\to\infty$; $x(t^+)= \lim_{t\to\tau_k^+}x(t)$, $x(t^-)=\lim_{t\to\tau_k^-} x(t)$. Several criteria on asymptotic stability are established using the method of Lyapunov functions. It is shown that system can be stabilized even if it contains no stable matrix $A$.

MSC:
93D20Asymptotic stability of control systems
93D30Scalar and vector Lyapunov functions
34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
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References:
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