# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020073

## 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

Received  July 2019 Revised  March 2020 Published  June 2020

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.

Citation: Rajesh Dhayal, Muslim Malik, Syed Abbas, Anil Kumar, Rathinasamy Sakthivel. Approximation theorems for controllability problem governed by fractional differential equation. Evolution Equations & Control Theory, doi: 10.3934/eect.2020073
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##### References:
Comparison between $z_{b}$ and $\overline{z}(b)$
Approximated optimal control $\overline{u}$
Numerical solution $\overline{z}$ corresponding to optimal control $\overline{u}$ with $\eta = 0.75$
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