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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:
 93B05 Controllability 34A37 Differential equations with impulses
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##### References:
 [1] Brockett, R. W.: Finite dimensional linear systems. (1970) · Zbl 0216.27401 [2] R. K. George, A class of admissible perturbations which preserves controllability of linear systems, submitted for publication. [3] Granas, A.: On a certain class of non-linear mappings in Banach spaces. Bull. acad. Polon. sci. 9, 867-871 (1957) · Zbl 0078.11701 [4] Joshi, M. C.; George, R. K.: Controllability of nonlinear systems. Numer. funct. Anal. optim. 10, 139-166 (1989) · Zbl 0676.93008 [5] Lakshmikantham, V.; Bainov, D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002 [6] Leela, S.; Mcrae, F. A.; Sivasundaram, S.: Controllability of impulsive differential equations. J. math. Anal. appl. 177, 24-30 (1993) · Zbl 0785.93016