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A convolution algebra of delay-differential operators and a related problem of finite spectrum assignability. (English) Zbl 0954.93007
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 $$s$$.

##### MSC:
 93B25 Algebraic methods 93B55 Pole and zero placement problems 93C23 Control/observation systems governed by functional-differential equations 93C80 Frequency-response methods in control theory
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