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

K. Ravikumar; Manil T. Mohan; A. Anguraj

doi:10.3934/naco.2020038

Article Contents

2021, Volume 11, Issue 3: 461-485. Doi: 10.3934/naco.2020038

This issue Previous Article The proximal methods for solving absolute value equation Next Article

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
^* Corresponding author: Manil T. Mohan

Received: April 2020

Revised: September 2020

Early access: September 17, 2020

Published online: September 17, 2020

The second author is supported by DST-INSPIRE Faculty Award (IFA17-MA110).

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K06; Secondary: 34A12, 37L05, 93B05.

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