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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), tτ k , x(t + )=x(t - )+C k x(t - ), t=τ k , where A,B,C k , k=1,2, are some constant matrices, r>0 is a delay constant, {τ k ,C k (x(τ k - )), k=1,2,} denotes the impulsive control law with 0<τ 1 <τ 2 <<τ k <τ k+1 <, τ k , as k; x(t + )=lim tτ k + x(t), x(t - )=lim tτ 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.

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