A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions

Xingyang Ye; Chuanju Xu; Xingyang Ye; Chuanju Xu

doi:10.3934/math.2021697

AIMS Mathematics

2021, Volume 6, Issue 11: 12028-12050. doi: 10.3934/math.2021697

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A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions

Xingyang Ye ¹,
Chuanju Xu ^{2
,
,}

1.
School of Science, Jimei University, 361021 Xiamen, China
2.
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing, Xiamen University, 361005 Xiamen, China

Received: 27 May 2021 Accepted: 12 August 2021 Published: 18 August 2021
MSC : 35R11, 49J20, 65K10, 65M32, 65M70, 65N15

In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.
- fractional optimal control problem,
- initial conditions,
- spectral method,
- a posteriori error
Citation: Xingyang Ye, Chuanju Xu. A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions[J]. AIMS Mathematics, 2021, 6(11): 12028-12050. doi: 10.3934/math.2021697

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Abstract

In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.

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