A priori error estimates for spectral Galerkin approximations of integral state-constrained fractional optimal control problems. (English) Zbl 1524.65902
Summary: 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.
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 65K10 | Numerical optimization and variational techniques |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
fractional optimal control problem; state constraint; spectral method; Jacobi polynomial; a priori error estimateReferences:
| [1] | O. P. AGRAWAL, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dyn., 38 (2004), pp. 323-337. · Zbl 1121.70019 |
| [2] | H. ANTIL AND E. OTÁROLA, A FEM for an optimal control problem of fractional powers of elliptic operators, SIAM J. Control Optim., 53 (2015), pp. 3432-3456. · Zbl 1331.49038 |
| [3] | H. ANTIL, E. OTÁROLA, AND A. J. SALGADO, A space-time fractional optimal control problem: analysis and discretization, SIAM J. Control Optim., 54 (2016), pp. 1295-1328. · Zbl 1339.49003 |
| [4] | I. BABUŠKA AND B. Q. GUO, Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces part I: approximability of functions in the weighted Besov spaces, SIAM J. Numer. Anal., 39 (2002), pp. 1512-1538. · Zbl 1008.65078 |
| [5] | A. H. BHRAWY AND M. A. ZAKY, An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations, Appl. Numer. Math., 111 (2017), pp. 197-218. · Zbl 1353.65106 |
| [6] | N. DU, H. WANG, AND W. B. LIU, A fast gradient projection method for a constrained fractional optimal control, J. Sci. Comput., 68 (2016), pp. 1-20. · Zbl 1344.65056 |
| [7] | B. Y. GUO, J. SHEN, AND L.-L. WANG, Generalized Jacobi polynomials/functions and their ap-plications, Appl. Numer. Math., 59 (2009), pp. 1011-1028. · Zbl 1171.33006 |
| [8] | B. Y. GUO AND L.-L. WANG, Jacobi approximations in non-uniformly Jacobi weighted Sobolev spaces, J. Approx. Theory, 128 (2004), pp. 1-41. · Zbl 1057.41003 |
| [9] | Z. P. HAO AND Z. Q. ZHANG, Optimal regularity and error estimate of a spectral galerkin method for fractional advection-diffusion-reaction equations, SIAM J. Numer. Anal., 58 (2020), pp. 211-233. · Zbl 1475.65121 |
| [10] | Z. P. HAO, G. LIN, AND Z. Q. ZHANG, Error estimates of a spectral Petrov-Galerkin method for two-sided fractional reaction-diffusion equations, Appl. Math. Comput., 374 (2020), 125045. · Zbl 1433.65211 |
| [11] | B. T. JIN, B. Y. LI, AND Z. ZHOU, Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint, IMA J. Numer. Anal., 40 (2020), pp. 377-404. · Zbl 1464.65065 |
| [12] | N. KUMAR AND M. MEHRA, Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost, Numer. Meth. Partial Differential Equations, 37 (2021), pp. 1693-1724. · Zbl 07776039 |
| [13] | S. Y. LI AND Z. J. ZHOU, Legendre pseudo-spectral method for optimal control problem governed by a time-fractional diffusion equation, Int. J. Comput. Math., 95 (2018), pp. 1308-1325. · Zbl 1499.65568 |
| [14] | S. Y. LI AND Z. J. ZHOU, Fractional spectral collocation method for optimal control problem gov-erned by space fractional diffusion equation, Appl. Math. Comput., 350 (2019), pp. 331-347. · Zbl 1428.49032 |
| [15] | X. LI AND C. J. XU, Existence and uniqueness of the weak solution of the space time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), pp. 1016-1051. · Zbl 1364.35424 |
| [16] | A. LOTFI, A development in the extended Ritz method for solving a general class of fractional varia-tional problems, J. Comput. Nonlin. Dyn., 15 (2020), 021001. |
| [17] | Z. MAO AND G. E. KARNIADAKIS, A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative, SIAM J. Numer. Anal., 56 (2018), pp. 24-49. · Zbl 1422.65428 |
| [18] | G. M. MOPHOU, Optimal control of fractional diffusion equation, Comput. Math. Appl., 61 (2011), pp. 68-78. · Zbl 1207.49006 |
| [19] | J. SHEN, T. TANG, AND L.-L. WANG, Spectral Methods: Algorithms, Analysis and Applica-tions, Springer, 2011. · Zbl 1227.65117 |
| [20] | X. J. TANG, Y. SHI, AND L.-L. WANG, A new framework for solving fractional optimal control problems using fractional pseudospectral methods, Automatica, 78 (2017), pp. 333-340. · Zbl 1357.49157 |
| [21] | N. TEMME, Orthogonal Polynomials, AMS Colloquium Publications, 1939. |
| [22] | F. Y. WANG, Z. Q. ZHANG, AND Z. J. ZHOU, A spectral Galerkin approximation of optimal control problem governed by fractional advection-diffusion-reaction equations, J. Comput. Appl. Math., 386 (2021), 113233. · Zbl 1456.49026 |
| [23] | X. Y. YE AND C. J. XU, A spectral method for optimal control problem governed by the abnormal diffusion equation with integral constraint on the state, Sci. Sin. Math., 46 (2016), pp. 1053-1070. · Zbl 1499.65584 |
| [24] | M. ZAKY AND J. MACHADO, On the formulation and numerical simulation of distributed-order fractional optimal control problems, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), pp. 177-189. · Zbl 1510.49018 |
| [25] | M. ZAYERNOURI, M. AINSWORTH, AND G. E. KARNIADAKIS, A unified Petrov-Galerkin spec-tral method for fractional PDEs, Comput. Method. Appl. M., 283 (2015), pp. 1545-1569. · Zbl 1425.65127 |
| [26] | M. ZAYERNOURI AND G. E. KARNIADAKIS, Discontinuous spectral element methods for time-and space-fractional advection equations, SIAM J. Sci. Comput., 36 (2014), pp. 684-707. · Zbl 1304.35757 |
| [27] | C. Y. ZHANG, H. P. LIU, AND Z. J. ZHOU, A priori error analysis for time-stepping discontinuous Galerkin finite element approximation of time fractional optimal control problem, J. Sci. Comput., 80 (2019), pp. 993-1018. · Zbl 1419.49028 |
| [28] | L. ZHANG AND Z. J. ZHOU, Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation, Appl. Numer. Math., 143 (2019), pp. 247-262. · Zbl 1425.49018 |
| [29] | Z. Q. ZHANG, Error estimate of spectral galerkin methods for a linear fractional reation-diffusion equation, J. Sci. Comput., 78 (2019), pp. 1087-1110. · Zbl 1462.65214 |
| [30] | Z. J. ZHOU, J. B. SONG, AND Y. P. CHEN, Finite element approximation of space fractional optimal control problem with integral state constraint, Numer. Math. Theor. Meth. Appl., 13 (2020), pp. 1027-1049. · Zbl 1474.65453 |
| [31] | Z. J. ZHOU AND Z. Y. TAN, Finite element approximation of optimal control problem governed by space fractional equation, J. Sci. Comput., 78 (2019), pp. 1840-1861. · Zbl 1417.49043 |
| [32] | Z. J. ZHOU AND C. Y. ZHANG, Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation, Numer. Algorithms, 79 (2018), pp. 437-455. · Zbl 1400.49035 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
