Controllability of nonlinear fractional evolution systems in Banach spaces: A survey

<p style='text-indent:20px;'>This paper presents a survey for some recent research on the controllability of nonlinear fractional evolution systems (FESs) in Banach spaces. The prime focus is exact controllability and approximate controllability of several types of FESs, which include the basic systems with classical initial and nonlocal conditions, FESs with time delay or impulsive effect. In addition, controllability results via resolvent operator are reviewed in detail. At last, the conclusions of this work and the research prospect are presented, which provides a reference for further study.</p>

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On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions

Open Mathematics ◽

10.1515/math-2020-0158 ◽

2020 ◽

Vol 18 (1) ◽

pp. 529-539

Author(s):

Xianghu Liu

Keyword(s):

Approximate Controllability ◽

Nonlocal Conditions ◽

Sufficient Conditions ◽

Fixed Point Problem ◽

Point Problem ◽

Evolution System ◽

Controlled Systems ◽

Uniqueness Of The Solution ◽

Fractional Evolution Systems ◽

Evolution Systems

Abstract The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness of the solution of the Hilfer fractional evolution systems. Using the variational approach, we obtain sufficient conditions for the finite approximate controllability of semilinear controlled systems. Finally, an example is provided to illustrate main results.

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Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions

Evolution Equations and Control Theory ◽

10.3934/eect.2017024 ◽

2017 ◽

Vol 6 (3) ◽

pp. 471-486 ◽

Cited By ~ 15

Author(s):

Jinrong Wang ◽

◽

Michal Fečkan ◽

Yong Zhou ◽

◽

...

Keyword(s):

Approximate Controllability ◽

Nonlocal Conditions ◽

Sobolev Type ◽

Fractional Evolution Systems ◽

Evolution Systems

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Existence Results for Fractional Evolution Systems with Riemann-Liouville Fractional Derivatives and Nonlocal Conditions

Fundamenta Informaticae ◽

10.3233/fi-2017-1506 ◽

2017 ◽

Vol 151 (1-4) ◽

pp. 487-504 ◽

Cited By ~ 1

Author(s):

P. Kalamani ◽

M. Mallika Arjunan ◽

D. Mallika ◽

D. Baleanu

Keyword(s):

Fractional Derivatives ◽

Nonlocal Conditions ◽

Existence Results ◽

Fractional Evolution Systems ◽

Evolution Systems

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

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0011 ◽

2020 ◽

Vol 23 (1) ◽

pp. 268-291 ◽

Cited By ~ 13

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.

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Finite-approximate controllability of fractional evolution equations: variational approach

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0050 ◽

2018 ◽

Vol 21 (4) ◽

pp. 919-936 ◽

Cited By ~ 5

Author(s):

Nazim I. Mahmudov

Keyword(s):

Hilbert Spaces ◽

Approximate Controllability ◽

Evolution Equations ◽

Nonlocal Conditions ◽

Sufficient Conditions ◽

Exact Controllability ◽

Heat Equations ◽

Natural Conditions ◽

Finite Dimensional ◽

Semilinear Evolution Equations

Abstract In this work we extend a variational method to study the approximate controllability and finite dimensional exact controllability (finite-approximate controllability) for the fractional semilinear evolution equations with nonlocal conditions in Hilbert spaces. Assuming the approximate controllability of the corresponding linear equation we obtain sufficient conditions for the finite-approximate controllability of the fractional semilinear evolution equation under natural conditions. The obtained results are generalization and continuation of the recent results on this issue. Applications to heat equations are treated.

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Exact controllability of fractional neutral integro-differential systems with state-dependent delay in Banach spaces

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.1515/auom-2016-0017 ◽

2016 ◽

Vol 24 (1) ◽

pp. 29-55 ◽

Cited By ~ 3

Author(s):

S. Kailasavalli ◽

D. Baleanu ◽

S. Suganya ◽

M. Mallika Arjunan

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Banach Spaces ◽

Point Theorem ◽

Resolvent Operator ◽

Exact Controllability ◽

Differential Systems ◽

State Dependent ◽

Banach Contraction ◽

State Dependent Delay

Abstract In this manuscript, we have a tendency to execute Banach contraction fixed point theorem combined with resolvent operator to analyze the exact controllability results for fractional neutral integro-differential systems (FNIDS) with state-dependent delay (SDD) in Banach spaces. An illustration is additionally offered to exhibit the achieved hypotheses.

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