Approximate Controllability and Existence of Mild Solutions for Riemann-Liouville Fractional Stochastic Evolution Equations with Nonlocal Conditions of Order 1 < α < 2

• Published in: Fractional calculus & applied analysis Vol. 22; no. 4; pp. 1086 - 1112
• Main Authors: Shu, Linxin, Shu, Xiao-Bao, Mao, Jianzhong
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 01.08.2019; De Gruyter; Nature Publishing Group
• Subjects: 26A33; 35R60; 47J35; Abstract Harmonic Analysis; Analysis; approximate controllability; Controllability; Evolution; Fixed points (mathematics); Functional Analysis; Integral Transforms; Mathematical analysis; Mathematics; mild solutions; Mönch fixed point theorem; noncompact measurement; Nonlinear control; Operational Calculus; Operators (mathematics); Primary 93B05; Research Paper; Riemann-Liouville fractional evolution equations; Secondary 34A08; sectorial operators; Stochastic systems; Theorems; mild solutions; approximate controllability; Mönch fixed point theorem; 35R60; Riemann-Liouville fractional evolution equations; Primary 93B05; 26A33; sectorial operators; noncompact measurement; Secondary 34A08; 47J35
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2019-0057

In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2. As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 < α < 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 < α < 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.