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Volume 15, Issue 3
A Priori Error Estimates for Spectral Galerkin Approximations of Integral State-Constrained Fractional Optimal Control Problems

Juan Zhang, Jiabin Song & Huanzhen Chen

Adv. Appl. Math. Mech., 15 (2023), pp. 568-582.

Published online: 2023-02

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  • Abstract

The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem, due to the global properties of fractional differential operators. In this paper, we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable. By the proposed first-order optimality condition consisting of a Lagrange multiplier, we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations. Furthermore, a priori error estimates for state, adjoint state and control variables are discussed in details. Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.

  • AMS Subject Headings

65N15, 65N35, 49J20

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COPYRIGHT: © Global Science Press

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@Article{AAMM-15-568, author = {Zhang , JuanSong , Jiabin and Chen , Huanzhen}, title = {A Priori Error Estimates for Spectral Galerkin Approximations of Integral State-Constrained Fractional Optimal Control Problems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2023}, volume = {15}, number = {3}, pages = {568--582}, abstract = {

The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem, due to the global properties of fractional differential operators. In this paper, we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable. By the proposed first-order optimality condition consisting of a Lagrange multiplier, we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations. Furthermore, a priori error estimates for state, adjoint state and control variables are discussed in details. Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0251}, url = {http://global-sci.org/intro/article_detail/aamm/21441.html} }
TY - JOUR T1 - A Priori Error Estimates for Spectral Galerkin Approximations of Integral State-Constrained Fractional Optimal Control Problems AU - Zhang , Juan AU - Song , Jiabin AU - Chen , Huanzhen JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 568 EP - 582 PY - 2023 DA - 2023/02 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2021-0251 UR - https://global-sci.org/intro/article_detail/aamm/21441.html KW - Fractional optimal control problem, state constraint, spectral method, Jacobi polynomial, a priori error estimate. AB -

The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem, due to the global properties of fractional differential operators. In this paper, we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable. By the proposed first-order optimality condition consisting of a Lagrange multiplier, we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations. Furthermore, a priori error estimates for state, adjoint state and control variables are discussed in details. Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.

Juan Zhang, Jiabin Song & Huanzhen Chen. (2023). A Priori Error Estimates for Spectral Galerkin Approximations of Integral State-Constrained Fractional Optimal Control Problems. Advances in Applied Mathematics and Mechanics. 15 (3). 568-582. doi:10.4208/aamm.OA-2021-0251
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