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A note on controllability of impulsive systems. (English) Zbl 0965.93015
This article investigates complete controllability of the control system with impulse effects $$x'(t)= A(t)x(t)+ B(t) u(t)+ f(t, x(t)),\quad t\ne t_k,\quad t\in [t_0,T],$$ $$x(t_k+)= (I+ D^ku(t_k)) x(t_k),\quad x(t_0)= x_0,$$ where, for each $t\in [t_0, T]$, the state $x(t)$ is an $n$-vector, the control $u(t)$ is an $m$-vector, $A(t)$ and $B(t)$ are $n\times n$ and $n\times m$ matrices, respectively, with piecewise continuous entries, and $0< t_1< t_2<\cdots< t_\rho< T$ are the time points when the impulsive controls $u(t_k)$ act. For each $k=1,2,\dots,\rho$, $D^ku(t_k)$ is an $n\times n$ diagonal matrix such that $$D^ku(t_k)= \sum^m_{i=1} d^k_i u_i(t_k)I,$$ where $I$ is the identity matrix on $\bbfR^n$ and $d^k_i\in \bbfR$.

MSC:
93B05Controllability
34A37Differential equations with impulses
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References:
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