A study on approximate controllability of non-autonomous evolution system with nonlocal conditions using sequence method

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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Solvability of control problem for fractional nonlinear differential inclusions with nonlocal conditions

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2019.4.2 ◽

2019 ◽

Vol 24 (4) ◽

Author(s):

Alka Chadha ◽

Rathinasamy Sakthivel ◽

Swaroop Nandan Bora

Keyword(s):

Differential Inclusions ◽

Approximate Controllability ◽

Measure Of Noncompactness ◽

Nonlocal Conditions ◽

Sufficient Conditions ◽

Fractional Differential Inclusions ◽

Multivalued Maps ◽

Nonlinear Differential Inclusions ◽

Approximately Controllable ◽

Fractional Differential

In this paper, we study the approximate controllability of nonlocal fractional differential inclusions involving the Caputo fractional derivative of order q ∈ (1,2) in a Hilbert space. Utilizing measure of noncompactness and multivalued fixed point strategy, a new set of sufficient conditions is obtained to ensure the approximate controllability of nonlocal fractional differential inclusions when the multivalued maps are convex. Precisely, the results are developed under the assumption that the corresponding linear system is approximately controllable.

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Approximate controllability for neutral impulsive differential inclusions with nonlocal conditions

Journal of Dynamical and Control Systems ◽

10.1007/s10883-011-9126-z ◽

2011 ◽

Vol 17 (3) ◽

pp. 359-386 ◽

Cited By ~ 16

Author(s):

Xianlong Fu

Keyword(s):

Differential Inclusions ◽

Approximate Controllability ◽

Nonlocal Conditions ◽

Impulsive Differential Inclusions

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Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0057 ◽

2019 ◽

Vol 22 (4) ◽

pp. 1086-1112 ◽

Cited By ~ 7

Author(s):

Linxin Shu ◽

Xiao-Bao Shu ◽

Jianzhong Mao

Keyword(s):

Approximate Controllability ◽

Stochastic Systems ◽

Evolution Equations ◽

Mild Solutions ◽

Nonlocal Conditions ◽

Stochastic Evolution ◽

Stochastic Evolution Equations ◽

Mönch Fixed Point Theorem ◽

Liouville Derivative ◽

Liouville Fractional Derivative

Abstract 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.

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