Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle

Arzu Ahmadova; Nazim I. Mahmudov; Juan J. Nieto

doi:10.3934/eect.2022008

Article Contents

2022, Volume 11, Issue 6: 1997-2015. Doi: 10.3934/eect.2022008

This issue Previous Article From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae Next Article Blow-up of solutions to a viscoelastic wave equation with nonlocal damping

Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle

1.
Department of Mathematics, Eastern Mediterranean University, Mersin 10, 99628, T.R. North Cyprus, Turkey
2.
Departamento de Estatística, Instituto de Matematicas, University of Santiago de Compostela, 15782, Spain

^*Corresponding author: Nazim I. Mahmudov
^*Corresponding author: Nazim I. Mahmudov

Received: August 31, 2021

Revised: November 30, 2021

Early access: February 2022

Published: December 2022

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Abstract

In this paper, we obtain a closed-form representation of a mild solution to the fractional stochastic degenerate evolution equation in a Hilbert space using the subordination principle and semigroup theory. We study aforesaid abstract fractional stochastic Cauchy problem with nonlinear state-dependent terms and show that if the Sobolev type resolvent families describing the linear part of the model are exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions for nonlinearity. We also establish conditions for stabilizability and prove that the stochastic nonlinear fractional Cauchy problem is exponentially stabilizable when the stabilizer acts linearly on the control systems. Finally, we provide applications to show the validity of our theory.

Keywords:

Fractional stochastic differential equations,
stochastic degenerate evolution equations,
semigroup theory,
subordination principle,
stability,
stabilization.

Mathematics Subject Classification: Primary: 26A33, 34G25, 44D03, 60H20, 93E15; Secondary: 35R11, 37H30,.

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