Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces

Arora, Sumit; Mohan, Manil T.; Dabas, Jaydev

doi:10.3934/mcrf.2020049

Cited by 9 publications

(5 citation statements)

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“…[30,46,47,50], etc). In the past two decades, a good number of publications discussed the problems of existence and approximate controllability of non-linear evolution systems (in Hilbert and Banach spaces), see for instance, [2,4,15,17,18,30,38,41], etc and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces

Arora¹,

Singh²,

Mohan³

et al. 2021

Preprint

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This article investigates the approximate controllability of second order nonautonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder's fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we verify our results to examine the approximate controllability of the non-autonomous wave equation with non-instantaneous impulses and finite delay in the application section.

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Section: Introductionmentioning

confidence: 99%

Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces

Arora¹,

Singh²,

Mohan³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…[29,52,53,66], etc. In the past two decades, the problem of approximate controllability of various kinds of systems (in Hilbert and Banach spaces) such as impulsive differential equations, functional differential equations, stochastic systems, Sobolev type evolution systems, etc, is extensively studied with the help of fixed point approach and produced excellent results, see for instance, [2,3,16,17,29,45], etc.…”

Section: Introductionmentioning

confidence: 99%

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Arora¹,

Mohan²,

Dabas³

2021

Preprint

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The present paper deals with the control problems governed by fractional noninstantaneous impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder's fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. We also provide an example to illustrate the efficiency of the developed results. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution and phase space, and approximate controllability of fractional order impulsive systems in Banach spaces.

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“…For the infinite dimensional control systems, the problem of approximate controllability is more substantial and is having broad range of applications, see for instance, [33,55,56,67], etc. In the past few decades, the problem of approximate controllability for different kinds of dynamical systems (in Hilbert and Banach spaces) such as fractional evolution equations, Sobolev type systems, delay (functional) differential equations, impulsive systems, stochastic differential equations, etc, has been widely studied by many researchers and they have produced excellent results, see for instance, [1,3,16,33,46,64] etc, and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Approximate controllability of non-instantaneous impulsive fractional evolution equations of order $1

Arora¹,

Mohan²,

Dabas³

2021

Preprint

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The current article examines the approximate controllability problem for noninstantaneous impulsive fractional evolution equations of order 1 < α < 2 with statedependent delay in separable reflexive Banach spaces. In order to establish sufficient conditions for the approximate controllability of our problem, we first formulate the linearregulator problem and obtain the optimal control in feedback form. By using this optimal control, we deduce the approximate controllability of the linear fractional control system of order 1 < α < 2. Further, we derive sufficient conditions for the approximate controllability of the nonlinear problem. Finally, we provide a concrete example to validate the efficiency of the derived results .

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Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces

Cited by 9 publications

References 50 publications

Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces

Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Approximate controllability of non-instantaneous impulsive fractional evolution equations of order $1

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