### AIMS Mathematics

2021, Issue 1: 979-997. doi: 10.3934/math.2021059
Research article

# Finite element approximation of time fractional optimal control problem with integral state constraint

• Received: 13 September 2020 Accepted: 27 October 2020 Published: 05 November 2020
• MSC : 65K10, 65N30

• In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

Citation: Jie Liu, Zhaojie Zhou. Finite element approximation of time fractional optimal control problem with integral state constraint[J]. AIMS Mathematics, 2021, 6(1): 979-997. doi: 10.3934/math.2021059

### Related Papers:

• In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

 [1] W. Liu, N. Yan, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math., 15 (2001), 285-309. doi: 10.1023/A:1014239012739 [2] W. Liu, N. Yan, A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40 (2003), 1850-1869. [3] E. Casas, J. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611. doi: 10.1137/050626600 [4] W. Gong, M. Hinze, Z. Zhou, Finite element method and a priori error estimates for Dirichlet boundary control problems governed by parabolic PDEs, J. Sci. Comput., 66 (2016), 941-967. doi: 10.1007/s10915-015-0051-2 [5] W. Liu, D. Yang, L. Yuan, C. Ma, Finite element approximation of an optimal control problem with integral state constraint, SIAM J. Numer. Anal., 48 (2011), 1163-1185. [6] D. Yang, et al., State-dependent switching control of delayed switched systems with stable and unstable modes, Math. Method Appl. Sci., 41 (2018), 6968-6983. doi: 10.1002/mma.5209 [7] D. Yang, X. Li, J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal-Hybri., 32 (2019), 294-305. doi: 10.1016/j.nahs.2019.01.006 [8] J. Bouchaud, A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Physics Reports, 195 (1990), 127-293. doi: 10.1016/0370-1573(90)90099-N [9] E. Barkai, R. Metzler, J. Klafter, Form continuous time random walks to the fractional Fokker-Planck equation, Physical Review E Statal Physics Plasmas Fluids & Related Interdiplinary Topics, 61 (2000), 132-138. [10] D. A. Benson, S. W. Wheatcraft, M. M. Meerschaeert, The fractional order governing equations of lévy motion, Water Resour. Res., 36 (2000), 1413-1423. doi: 10.1029/2000WR900032 [11] M. M. Meerschaert, A. Sikorskii, Stochastic models for fractional calculus, 2012. [12] Y. N. Zhang, Z. Z. Sun, H. L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys., 265 (2014), 195-210. doi: 10.1016/j.jcp.2014.02.008 [13] H. L. Qiao, Z. G. Liu, A. J. Cheng, Two unconditionally stable difference schemes for time distributed-order differential equation based on Caputo-Fabrizio fractional derivative, Adv. Differ. Equ., 2020 (2020), 1-17. doi: 10.1186/s13662-019-2438-0 [14] V. Ervin, J. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. Part. D. E., 22 (2010), 558-576. [15] B. Jin, R. Lazarov, Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput., 38 (2016), A146-A170. doi: 10.1137/140979563 [16] H. Chen, H. Wang, Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation, J. Comput. Appl. Math., 296 (2016), 480-498. doi: 10.1016/j.cam.2015.09.022 [17] B. Li, H. Luo, X. Xie, Analysis of a time-stepping scheme for time fractional diffusion problems with nonsmooth data, SIAM J. Numer. Anal., 57 (2019), 779-798. doi: 10.1137/18M118414X [18] Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552. doi: 10.1016/j.jcp.2007.02.001 [19] B. Li, H. Luo, X. Xie, A time-spectral algorithm for fractional wave problems, J. Sci. Comput., 7 (2018), 1164-1184. [20] Z. Zhou, W. Gong, Finite element approximation of optimal control problems governed by time fractional diffusion equation, Comput. Math. Appl., 71 (2016), 301-318. doi: 10.1016/j.camwa.2015.11.014 [21] Z. Zhou, C. Zhang, Time-stepping discontinuous Galerkin approximation of optimal control problem governed by time fractional diffusion equation, Numer. Algorithms, 79 (2018), 437-455. doi: 10.1007/s11075-017-0445-3 [22] M. Gunzburger, J. Wang, Error analysis of fully discrete finite element approximations to an optimal control problem governed by a time-fractional PDE, SIAM J. Control Optim., 57 (2019), 241-263. doi: 10.1137/17M1155636 [23] B. Jin, B. Li, Z. Zhou, Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint, IMA J. Numer. Anal., 40 (2020), 377-404. doi: 10.1093/imanum/dry064 [24] C. Zhang, H. Liu, Z. Zhou, A priori error analysis for time-stepping discontinuous Galerkin finite element approximation of time fractional optimal control problem, J. Sci. Comput., 80 (2019), 993-1018. doi: 10.1007/s10915-019-00964-9 [25] X. Ye, C. Xu, Spectral optimization methods for the time fractional diffusion inverse problem, Numer. Math. Theory Me., 6 (2013), 499-519. doi: 10.4208/nmtma.2013.1207nm [26] L. Zhang, Z. Zhou, Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation, Appl. Numer. Math., 143 (2019), 247-262. doi: 10.1016/j.apnum.2019.04.003 [27] Z. Zhou, Z. Tan, Finite element approximation of optimal control problem governed by space fractional equation, J. Sci. Comput., 78 (2019), 1840-1861. doi: 10.1007/s10915-018-0829-0 [28] X. Ye, C. Xu, A spectral method for optimal control problem governed by the abnormal diffusion equation with integral constraint on the state, Chinese science: Mathematics, 7 (2016), 1053-1070. [29] H. Antil, D. Verma, M. Warma, Optimal control of fractional elliptic PDEs with state constraints and characterization of the dual of fractional order Sobolev spaces, 186 (2020), 1-23. [30] Z. Zhou, J. Song, Y. Chen, Finite element approximation of space fractional optimal control problem with integral state constraint, Numer. Math. Theory Me., 13 (2020), 1027-1049. doi: 10.4208/nmtma.OA-2019-0201 [31] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, 1984. [32] E. Cacas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327. doi: 10.1137/S0363012995283637 [33] E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM. Contr. Optim. Ca., 8 (2002), 345-374. doi: 10.1051/cocv:2002049
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

2.2 3

Article outline

Tables(8)

• On This Site