A note on controllability of impulsive systems
This article investigates complete controllability of the control system with impulse effects \[ x'(t)= A(t)x(t)+ B(t) u(t)+ f(t, x(t)),\quad t\neq t_k,\quad t\in [t_0,T], \] \[ x(t_k+)= (I+ D^ku(t_k)) x(t_k),\quad x(t_0)= x_0, \] 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 \(\mathbb{R}^n\) and \(d^k_i\in \mathbb{R}\).
scientific article; zbMATH DE number 1423863