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Analysis of fractional delay systems of retarded and neutral type. (English) Zbl 1007.93065

The authors study fractional systems with scalar transfer function given by

$P\left(s\right)=\frac{{\sum }_{i=1}^{{n}_{2}}{q}_{i}\left(s\right){e}^{-{\beta }_{i}s}}{{\sum }_{i=1}^{{n}_{1}}{p}_{i}\left(s\right){e}^{-{\gamma }_{i}s}},$

where $0={\gamma }_{0}<{\gamma }_{1}<\cdots <{\gamma }_{{n}_{1}}$, $0={\beta }_{0}<{\beta }_{1}<\cdots <{\beta }_{{n}_{2}}$, the ${p}_{i}$ being polynomials of the form $\sum {a}_{k}{s}^{{\alpha }_{k}}$ with ${\alpha }_{k}\in {ℝ}_{+}$ and the ${q}_{i}$ being polynomials of the form $\sum {b}_{k}{s}^{{\delta }_{k}}$ with ${\delta }_{k}\in {ℝ}_{+}$. In order to analyse these systems, a frequency-domain approach is taken. The first main results provide a BIBO-stability analysis, which is in general quite difficult to perform since the impulse response of such a system cannot usually be written down explicitly, and the transfer function has a branch point on the imaginary axis. In particular, the BIBO stability of retarded and neutral fractional systems is related to the location of their poles. Further, sufficient conditions for nuclearity are given.

##### MSC:
 93D25 Input-output approaches to stability of control systems 93C23 Systems governed by functional-differential equations 47B25 Symmetric and selfadjoint operators (unbounded) 93C80 Frequency-response methods