scholarly journals APPROXIMATE CONTROLLABILITY OF SOBOLEV TYPE FRACTIONAL EVOLUTION EQUATIONS OF ORDER $\alpha\in (1,2)$ VIA RESOLVENT OPERATORS

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
He Yang ◽  
Keyword(s):  
Approximate Controllability ◽  
Evolution Equations ◽  
Sobolev Type ◽  
Resolvent Operators ◽  
Fractional Evolution Equations

Existence and approximate controllability of fractional evolution equations with nonlocal conditions via resolvent operators

2020 ◽  
Vol 23 (1) ◽  
pp. 268-291 ◽  
Author(s):  
Pengyu Chen ◽  
Xuping Zhang ◽  
Yongxiang Li
Keyword(s):  
Resolvent Operator ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Control Function ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Resolvent Operators ◽  
Fractional Evolution Equations ◽  
Theoretical Results

AbstractIn this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.

scholarly journals A study on controllability of impulsive fractional evolution equations via resolvent operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Resolvent Operator ◽  
Evolution Equations ◽  
Validity Study ◽  
Resolvent Operators ◽  
Fractional Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Alpha Beta

AbstractIn this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ ( α , β ) -resolvent operator, we concern with the term $u'(\cdot )$ u ′ ( ⋅ ) and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ u ( b ) = u b and $u'(b)=u'_{b}$ u ′ ( b ) = u b ′ . Finally, we present an application to support the validity study.

scholarly journals Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
N. I. Mahmudov
Keyword(s):  
Fixed Point ◽  
Linear System ◽  
Fixed Point Theorem ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Sobolev Type ◽  
Complete Controllability ◽  
Approximately Controllable ◽  
Sobolev Type Differential Equations

We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.

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