Criteria for Function Space Controllability of Linear Neutral Systems

Necessary and sufficient conditions for the exact state controllability of the linear autonomous differential difference equation of neutral type, $\dot x(t) = A_{ - 1} \dot x(t - h) + A_0 x(t) + A_1 x(t - h) + Bu(t)$, are given for the Sobolev state space $W_2^{(1)} ([ - h,0],R^n )$. In particular when B is an $n \times 1$ matrix, it is shown that the controllability of the above n-dimensional system on the interval $[0,\tau ]$, $\tau > nh$, is equivalent to rank $[B,A_{ - 1} B, \cdots ,A_{ - 1}^{n - 1} B] = n$ and that a certain two point boundary value problem for a related homogeneous ordinary differential equation have only the trivial solution. Practical criteria based thereon entail only elementary computations involving the coefficient matrices $[A_{ - 1} ,A_0 ,A_1 ,B]$ but these computations can be tedious when $n > 3$. The condition that the two point boundary value problem have only the trivial solution is often equivalent to a much simpler condition: $K(\lambda )\mathcal{S}_\lambda ^n \ne 0$ f...