## Abstract

In this work, by theory of analytic semi-groups, fractional powers of operators, and resolvent operator theory, we study the approximate controllability of a semi-linear neutral integro-differential equation with nonlocal conditions. Under the assumption of controllability on the corresponding linear system, we obtain the sufficient conditions for the considered semi-linear integro-differential system. In particular, the compactness condition or Lipschitz condition for the function *g* in the nonlocal condition appearing in literature is not required here. An example is also provided to illustrate the application of the obtained results.

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## Acknowledgements

We would like to thank the referees greatly for the valuable comments and suggestions to this paper.

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This work is supported by NSF of China (Nos. 11671142 and 11771075), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000).

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Huang, H., Fu, X. Approximate Controllability of Semi-Linear Neutral Integro-Differential Equations with Nonlocal Conditions.
*J Dyn Control Syst* **26**, 127–147 (2020). https://doi.org/10.1007/s10883-019-09438-5

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DOI: https://doi.org/10.1007/s10883-019-09438-5