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Algebraic interpretations of the spectral controllability of a linear delay system. (English) Zbl 0891.93014
The approach used in this paper is the one introduced by M. Fliess [Syst. Control Lett. 15, 391-396 (1990; Zbl 0727.93024) and Linear Alg. Appl. 203-204, 429-442 (1994; Zbl 0802.93010)]. A system is thus represented as a finitely generated module over the ring of exponential polynomials $${\mathbb R}[s, e^{-{\mathbf h}s}] \widehat {=} {\mathbb R}[s, e^{-h_{1}s} , \ldots e^{-h_{r}s}]$$ (where $$s$$ is the Laplace variable). The purpose of this paper is to study spectral controllability of a delay system following this approach.
Three general controllability definitions are given on an $${\mathbb R} [ {{d} \over {dt}}, {\pmb\delta} ]$$-algebra $$A$$, viz. $$A$$-free, $$A$$-projective and $$A$$-torsion free controllability. By varying the algebra $$A$$ in these three definitions, most of the controllability notions in the literature are found back.
Spectral controllability of a delay system can be characterized as the emptiness of the variety associated with the Fitting ideal of the module. Let $${\mathfrak S}_{r}$$ be the ring of localized and distributed delay differential operators: $${\mathfrak S}_{r} = {\mathbb R}(s)[e^{-{\mathbf h}s},e^{{\mathbf h}s}] \cap {\mathfrak F}$$ where $${\mathfrak F}$$ denotes the ring of entire functions. The main result states that if $${\mathbb R}[s,e^{-{\mathbf h}s},e^{{\mathbf h}s}] \otimes_{{\mathbb R}[s, e^{-{\mathbf h}s}]} \Lambda$$ is torsion free, then $$\Lambda$$ is spectrally controllable if and only if $${\mathfrak S}_{r} \otimes_{{\mathbb R}[s, e^{-{\mathbf h}s}]} \Lambda$$ is torsion free.
The final part of the paper recasts existing characterizations into this setting and gives a behavioral interpretation.

##### MSC:
 93B05 Controllability 93B25 Algebraic methods 34K35 Control problems for functional-differential equations
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