Criteria for Function Space Controllability of Linear Neutral Systems

Abstract

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: $K(\lambda )\mathcal{S}_\lambda ^n \ne 0$ for all complex $\lambda $, where $\mathcal{S}_\lambda ^n = [1,e^{ - \lambda h} , \cdots ,e^{ - (n - 1)\lambda h} ]^T $ and $K(\lambda )$ is an $n \times n$ matric polynomial of degree $n - 1$ which is constructed from the matrix $[A_{ - 1} ,A_0 ,A_1 ,B]$. This equivalence for the general case is still an open question. It is shown that the collection of controllable neutral systems form an open, dense subset of the collection of all neutral systems of the type considered. This is in marked contrast with the situation for retarded systems. It is also proved (for general B) that when the matrix, $[B,A_{ - 1} B, \cdots ,A_{ - 1}^{n - 1} B]$, has rank n, the solution operator, $u \to x_\tau ( \cdot ,0,u)$, for quite general neutral systems has closed range and finite deficiency. This often turns out to be an adequate substitute for a controllability assumption.