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doi: 10.3934/naco.2020038

## Approximate controllability of a non-autonomous evolution equation in Banach spaces

 1 Department of Mathematics, PSG College of Arts and Science, Coimbatore, Tamil Nadu 641 046, INDIA 2 Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

* Corresponding author: Manil T. Mohan

Received  April 2020 Revised  September 2020 Published  September 2020

Fund Project: The second author is supported by DST-INSPIRE Faculty Award (IFA17-MA110)

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained.

Citation: K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020038
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