A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability
A well-known algebraic model for time-delay systems with commensurate delays is used. Namely, each polynomial \[ p=\sum^L_{j=0} \sum^N_{i=0}p_{ij}s^iz^j \in\mathbb{R}[s,z] \] of two variables with real coefficients gives rise to the delay-differential operator \[ \left(p\left ({d\over dt}, \sigma \right) w\right)(t)= \sum^L_{j=0} \sum^N_{i=0} p_{ij}w^{(i)} (t-j), \] which is considered as an operator on the space of real valued infinitely differentiable functions defined on \(\mathbb{R}\). Within this model, the ring \[ {\mathcal H}= \left\{{p \over r}:p,r\in \mathbb{R}[s,z],\;r\neq 0,\text{ Ker} r\left({d\over dt}, \sigma\right)\subseteq\text{Ker} p\left({d\over dt},\sigma \right) \right \} \] and its various subrings (in which no polynomials in \(z^{-1}\) are allowed) are studied. It was proved in a previous paper by the author that \({\mathcal H}\) is a Bézout domain. In the reviewed paper, it is shown that the ring \({\mathcal H}\) and its subrings can be interpreted as algebras of distributions. In the last section, the problem of coefficient assignability via dynamic state feedback of systems of the form \(\dot x=A(\sigma)x+ B(\sigma) u\) is discussed. Here \(A(\sigma)\) and \(B(\sigma)\) are point delay matrices of sizes \(n\times n\) and \(n\times m\), respectively. It is proved that the system is coefficient assignable (in a weak sense) if and only if it is spectrally controllable, i.e., \[ \text{rank} \bigl[sI-A(e^{-s}),\;B(e^{-s})\bigr]= n \] for all complex numbers \(s\).
scientific article; zbMATH DE number 1428465