Open AccessExponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle
Author(s) -
Arzu Ahmadova,
Nazım I. Mahmudov,
Juan J. Nieto
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.2022008
Subject(s) - mathematics , lipschitz continuity , hilbert space , sobolev space , degenerate energy levels , semigroup , mathematical analysis , nonlinear system , cauchy distribution , initial value problem , exponential stability , resolvent , pure mathematics , physics , quantum mechanics
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.