Modal control of systems with delay (Q2739949)

The authors consider a system of differential equations with delay NEWLINE\[NEWLINE\dot x(t)=Ax(t)+A_1x(t-\tau)+bu(t),NEWLINE\]NEWLINE where \(A=\{a_{ij}\}\), \(A_1= \{a_{ij}^1\}\), \(i,j=1,\ldots,n,\;b\in R^{n}\). The problem of modal control for the considered system is the following. Find a control in the form \(u(t)=\sum_{j=0}^{m}c_{j}^{T}x(t-j\tau)\), \(c_{j}^{T}= (c_{1j},\ldots, c_{nj})\), such that the characteristic equation of the closed system NEWLINE\[NEWLINE\dot x(t)=Ax(t)+A_1x(t-\tau)+b\sum_{j=0}^{m} c_{j}^{T}x(t-j\tau)NEWLINE\]NEWLINE has a given form \(\lambda^{n}+q_1(\lambda) \lambda^{n-1}+\ldots+q_{n}(\lambda)=0\), \(q_{i}=\sum_{j=0}^{i} e^{-i\lambda\tau}q_{ij}, q_{00}=1\). The necessary and sufficient conditions of modal controllability of the considered system are obtained.