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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.

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