Controllability of singular control systems with delay (Q2702905)

The authors deal with the controllability of singular control systems with delay. Several necessary and sufficient conditions for the controllability are given, which are similar to those for regular control systems. For example, one of them is that the system \(N\dot x(t)= x(t)+ Bx(t-1)+ Cu(t)\), where \(x\in\mathbb{R}^n\), \(B\in \mathbb{R}^{n\times n}\), \(C\in \mathbb{R}^{n\times m}\), \(u\in \mathbb{R}^m\) is completely controllable for \(t\in [t_0+ k,t_0+ k+1)\), \(k\) is an integer, if and only if the matrix \(J_k= (A_0,A_1,\dots, A_k)\), where NEWLINE\[NEWLINEA_n= (B_{j,0}C, B_{j,1}C,\dots, B_{j,(j+1)(j-1)}C),\quad B_{j,n}= \sum^n_{k=0} N^k B_{j-1,n-k},NEWLINE\]NEWLINE has full rank.