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Approximate Controllability of Fractional Differential Equations with State-Dependent Delay

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Abstract

The systems governed by delay differential equations come up in different fields of science and engineering but often demand the use of non-constant or state-dependent delays. The corresponding model equation is a delay differential equation with state-dependent delay as opposed to the standard models with constant delay. The concept of controllability plays an important role in physics and mathematics. In this paper, first we study the approximate controllability for a class of nonlinear fractional differential equations with state-dependent delays. Then, the result is extended to study the approximate controllability fractional systems with state-dependent delays and resolvent operators. A set of sufficient conditions are established to obtain the required result by employing semigroup theory, fixed point technique and fractional calculus. In particular, the approximate controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is approximately controllable. Also, an example is presented to illustrate the applicability of the obtained theory.

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Authors and Affiliations

  1. Department of Mathematics, Sungkyunkwan University, Suwon, 440-746, South Korea

    Sakthivel Rathinasamy

  2. Department of Mathematics, Anhui Normal University, Wuhu, 241000, China

    Ren Yong

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  1. Sakthivel Rathinasamy
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  2. Ren Yong
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Correspondence to Ren Yong.

Additional information

The work of R. Sakthivel is supported by the Korean Research Foundation Grant funded by the Korean Government with Grant number KRF 2011-0005449.

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Rathinasamy, S., Yong, R. Approximate Controllability of Fractional Differential Equations with State-Dependent Delay. Results. Math. 63, 949–963 (2013). https://doi.org/10.1007/s00025-012-0245-y

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