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Approximate Controllability of Stochastic Degenerate Evolution Equations: Decomposition of a Hilbert Space

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Abstract

We study a class of dynamic control systems described by nonlinear fractional stochastic degenerate evolution equations in Hilbert spaces. We investigate the possibility of replacing a degenerate Cauchy problem by an equivalent problem in the factor space \({\mathcal {H}}^{\bot }={\mathcal {H}}\setminus \text {Ker}(L)\). The core of this paper is to factorize the degenerate Cauchy problem, to derive a mild solution of stochastic system based on the perturbation theory for linear operators and to study approximate controllability results. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic degenerate evolution equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic degenerate control system under the assumption that the corresponding linear system is approximately controllable. As an application, the abstract results are illustrated by the application in complex media electromagnetic. In addition, the fractional stochastic partial differential equations (fractional SPDEs) are discussed and approximate controllability results are proved by verifying main assumptions.

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Acknowledgements

The author thanks Professor Nazim Mahmudov for useful discussions and referees for their comments that help improve the quality of the paper.

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  1. Faculty of Mathematics, University of Duisburg-Essen, Essen, 45127, Germany

    Arzu Ahmadova

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  1. Arzu Ahmadova
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Correspondence to Arzu Ahmadova.

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Ahmadova, A. Approximate Controllability of Stochastic Degenerate Evolution Equations: Decomposition of a Hilbert Space. Differ Equ Dyn Syst (2023). https://doi.org/10.1007/s12591-023-00631-4

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