Geodesic
Issues of unique solvability and approximate controllability of linear fractional order equations with a H\"olderian right-hand side
Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 1, pp. 5-21

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Issues of unique solvability and approximate controllability of linear fractional order evolution equations, both resolved with respect to the Riemann — Liouville fractional derivative (nondegenerate) and containing an irreversible operator at it (degenerate), are investigated. It is assumed that an operator on the right side of a non-degenerate equation or a pair of operators in a degenerate equation generates an analytic in a sector resolving family of operators of the corresponding homogeneous equation. New results on the solvability of inhomogeneous equations of such classes with a Hölder continuous function on the right side are obtained. These results allow us to find criteria for the approximate controllability of a degenerate system in fixed time, in free time, and in the case of systems with finite-dimensional input. The initial state of the degenerate control system is set by the Showalter — Sidorov type conditions. Based on the obtained abstract results, we found a criterion for the approximate controllability of a distributed control system, the dynamics of which is described by the linearized system of Navier — Stokes equations of fractional order in time.
Keywords: fractional Riemann — Liouville derivative, analytic in a sector resolving family of operators, degenerate evolution equation, Hölder condition, approximate controllability. 
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     author = {A. S. Avilovich and D. M. Gordievskikh and V. E. Fedorov},
     title = {Issues of unique solvability and approximate controllability of linear fractional order equations with a {H\"olderian} right-hand side},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
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     publisher = {mathdoc},
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     number = {1},
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     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_1_a0/}
}
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A. S. Avilovich; D. M. Gordievskikh; V. E. Fedorov. Issues of unique solvability and approximate controllability of linear fractional order equations with a H\"olderian right-hand side. Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 1, pp. 5-21. http://geodesic.mathdoc.fr/item/CHFMJ_2020_5_1_a0/