Approximate controllability of fractional nonlinear differential inclusions

Sakthivel, R.; Ganesh, R.; Anthoni, S. Marshal

doi:10.1016/j.amc.2013.09.068

Cited by 112 publications

(80 citation statements)

References 32 publications

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“…The nonlocal term g has a better effect on the solution and is more precise for physical measurements than the classical condition x(0) = x 0 alone [38]. For example, g(x) can be written as…”

Section: Preliminariesmentioning

confidence: 99%

“…For fractional differential inclusions, Sakthivel et al [38] formulated and proved a new set of sufficient conditions for the approximate controllability of fractional nonlinear differential inclusions. Yan and Jia [43] investigated the existence of mild solutions for a class of impulsive fractional partial neutral functional integrodifferential inclusions with infinite delay and analytic α-resolvent operators in Banach spaces.…”

mentioning

confidence: 99%

“…Moreover, approximate controllable systems are more prevalent and very often approximate controllability is completely adequate in applications (see [37,38]). Approximate controllability for semilinear deterministic and stochastic control systems can be found in Mahmudov [26].…”

mentioning

confidence: 99%

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Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

Slama

Boudaoui

2017

Arab. J. Math.

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In this paper we consider a class of fractional nonlinear neutral stochastic evolution inclusions with nonlocal initial conditions in Hilbert space. Using fractional calculus, stochastic analysis theory, operator semigroups and Bohnenblust-Karlin's fixed point theorem, a new set of sufficient conditions are formulated and proved for the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions under the assumption that the associated linear part of the system is approximately controllable. An example is provided to illustrate the theory. Mathematics Subject Classification

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Section: Preliminariesmentioning

confidence: 99%

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confidence: 99%

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Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

Slama

Boudaoui

2017

Arab. J. Math.

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“…Another new operator called conformable fractional derivative has some properties that are distinct from those usual in other formulations [2]. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have many results (see for example [3,7,19,26,[32][33][34][35]). On the other hand, Hilfer [13] proposed a generalized Riemann-Liouville fractional derivative, for short, Hilfer fractional derivative, which includes Riemann-Liouville fractional derivative and Caputo fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, it is important, in fact necessary to study the weaker concept of controllability, namely approximate controllability for nonlinear systems. In recent years, there are some papers on the approximate controllability of the nonlinear evolution systems under different conditions [7,19,21,25,26,32]. The conditions are established with the help of semigroup theory and fixed point theorem under the assumption that the associated linear system is approximately controllable.…”

Section: Introductionmentioning

confidence: 99%

Approximate controllability of impulsive Hilfer fractional differential inclusions

Du¹,

Jiang²,

Niazi³

2017

J. Nonlinear Sci. Appl.

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In this paper, firstly by utilizing the theory of operators semigroup, probability density functions via impulsive conditions, we establish a new PC 1−ν -mild solution for impulsive Hilfer fractional differential inclusions. Secondly we prove the existence of mild solutions for the impulsive Hilfer fractional differential inclusions by using fractional calculus, multi-valued analysis and the fixed-point technique. Then under some assumptions, the approximate controllability of associated system are formulated and proved. An example is provided to illustrate the application of the obtained theory.

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Existence and Controllability Results for a New Class of Impulsive Stochastic Partial Integro‐Differential Inclusions with State‐Dependent Delay

Zhang

Jia

2016

Asian Journal of Control

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In this paper, we study a new class of impulsive stochastic partial integro‐differential inclusions with state‐dependent delay in separable Hilbert spaces. Firstly, by using stochastic analysis theory, analytic resolvent operators, fractional powers of closed operators and suitable fixed point theorems, we prove the existence of mild and extremal mild solutions for these systems in the α‐norm. Secondly, we establish the controllability of the controlled stochastic partial integro‐differential inclusions with not instantaneous impulses. The results are obtained under the mixed Lipschitz and Carathéodory conditions. Finally, an example is provided to show the application of our results.

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Approximate controllability of fractional nonlinear differential inclusions

Cited by 112 publications

References 32 publications

Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

Approximate controllability of impulsive Hilfer fractional differential inclusions

Existence and Controllability Results for a New Class of Impulsive Stochastic Partial Integro‐Differential Inclusions with State‐Dependent Delay

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