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A new approach on the approximate controllability of fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces

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  • Raja, M. Mohan
  • Vijayakumar, V.
  • Udhayakumar, R.
  • Zhou, Yong

Abstract

This manuscript is mainly focusing on approximate controllability for fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces. We consider a class of control systems governed by the fractional differential evolution equations. By using the results on fractional calculus, cosine and sine functions of operators, and Schauder’s fixed point theorem, a new set of sufficient conditions are formulated which guarantees the approximate controllability of fractional differential evolution systems. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Lastly, we present theoretical and practical applications to support the validity of the study.

Suggested Citation

  • Raja, M. Mohan & Vijayakumar, V. & Udhayakumar, R. & Zhou, Yong, 2020. "A new approach on the approximate controllability of fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    • Handle: RePEc:eee:chsofr:v:141:y:2020:i:c:s0960077920307062
    DOI: 10.1016/j.chaos.2020.110310
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    References listed on IDEAS

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    1. Ludwik Byszewski & Haydar Akca, 1997. "On a mild solution of a semilinear functional-differential evolution nonlocal problem," International Journal of Stochastic Analysis, Hindawi, vol. 10, pages 1-7, January.
    2. Kavitha, K. & Vijayakumar, V. & Udhayakumar, R., 2020. "Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    3. Vijayakumar, V. & Udhayakumar, R., 2020. "Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
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    Cited by:

    1. Shukla, Anurag & Vijayakumar, V. & Nisar, Kottakkaran Sooppy, 2022. "A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order r∈(1,2)," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    2. Chen, Weihao & Liu, Yansheng & Zhao, Daliang, 2024. "Approximate controllability for a class of stochastic impulsive evolution system with infinite delay involving the fractional substantial derivative," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    3. Sumit Arora & Manil T. Mohan & Jaydev Dabas, 2023. "Finite-Approximate Controllability of Impulsive Fractional Functional Evolution Equations of Order $$1," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 855-890, June.
    4. Boudjerida, Assia & Seba, Djamila, 2021. "Approximate controllability of hybrid Hilfer fractional differential inclusions with non-instantaneous impulses," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    5. Dineshkumar, C. & Udhayakumar, R. & Vijayakumar, V. & Nisar, Kottakkaran Sooppy & Shukla, Anurag, 2021. "A note on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order 1," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1003-1026.
    6. Singh, Ajeet & Shukla, Anurag & Vijayakumar, V. & Udhayakumar, R., 2021. "Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    7. Daliang Zhao, 2023. "Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay," Mathematics, MDPI, vol. 11(19), pages 1-19, September.

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