Issues of unique solvability and approximate controllability of linear fractional order equations with a H\"olderian right-hand side

A. S. Avilovich ; D. M. Gordievskikh ; V. E. Fedorov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 5 (2020) no. 1, pp. 5-21.

Résumé

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.

Détail

Keywords: fractional Riemann — Liouville derivative, analytic in a sector resolving family of operators, degenerate evolution equation, Hölder condition, approximate controllability.

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