Approximation theorems for controllability problem governed by fractional differential equation

Rajesh Dhayal; Muslim Malik; Syed Abbas; Anil Kumar; Rathinasamy Sakthivel

doi:10.3934/eect.2020073

Article Contents

2021, Volume 10, Issue 2: 411-429. Doi: 10.3934/eect.2020073

This issue Previous Article Well-posedness of infinite-dimensional non-autonomous passive boundary control systems Next Article

Approximation theorems for controllability problem governed by fractional differential equation

1.
School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175 005, India
2.
Department of Mathematics, BITS Pilani KK Birla Goa Campus, Zuarinagar, Goa-403 726, India
3.
Department of Applied Mathematics, Bharathiar University, Coimbatore, Tamilnadu-641 046, India

^* Corresponding author: Rathinasamy Sakthivel
^* Corresponding author: Rathinasamy Sakthivel

Received: June 30, 2019

Revised: February 29, 2020

Early access: June 2020

Published: June 2021

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Abstract

In this manuscript, we discuss the optimal control problem for a nonlinear system governed by the fractional differential equation in a separable Hilbert space $ X $. We utilize the fixed point technique and $ \eta $-resolvent family to present the existence of control for the fractional system. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem. Further, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Finally, the main results are validated with the aid of an example.

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Mathematics Subject Classification: Primary: 34A08; Secondary: 93B05, 49J20, 93C25.

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Figure 1. Comparison between $ z_{b} $ and $ \overline{z}(b) $

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Figure 2. Approximated optimal control $ \overline{u} $

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Figure 3. Numerical solution $ \overline{z} $ corresponding to optimal control $ \overline{u} $ with $ \eta = 0.75 $

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