Open AccessOn analytic semigroup generators involving Caputo fractional derivative
Author(s) -
Piotr Grabowski
Publication year - 2022
Publication title -
evolution equations and control theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 19
eISSN - 2163-2480
pISSN - 2163-2472
DOI - 10.3934/eect.2022014
Subject(s) - semigroup , mathematics , fractional calculus , generator (circuit theory) , operator (biology) , sobolev space , pure mathematics , analytic semigroup , derivative (finance) , perturbation (astronomy) , order (exchange) , mathematical analysis , physics , power (physics) , biochemistry , chemistry , finance , repressor , quantum mechanics , transcription factor , financial economics , economics , gene
Our investigations are motivated by the well - posedness problem of some dynamical models with anomalous diffusion described by the Caputo spatial fractional derivative of order \begin{document}$ \alpha \in (1, 2) $\end{document} . We propose a characterization of an exponentially stable analytic semigroup generator using the inverse operator. This characterization enables us to establish the form of a generator involving the Caputo fractional derivative, under various boundary conditions. In particular, the results simplify those known from literature obtained by means of the fractional Sobolev spaces and some perturbation results. Going further, we show how to construct a control system in factor form, having such a generator as the state operator.