A space-time spectral method for the time fractional diffusion optimal control problems
- Xingyang Ye^{1} and
- Chuanju Xu^{2}Email author
https://doi.org/10.1186/s13662-015-0489-4
© Ye and Xu; licensee Springer. 2015
Received: 8 January 2015
Accepted: 27 April 2015
Published: 15 May 2015
Abstract
In this paper, we study the Galerkin spectral approximation to an unconstrained convex distributed optimal control problem governed by the time fractional diffusion equation. We construct a suitable weak formulation, study its well-posedness, and design a Galerkin spectral method for its numerical solution. The contribution of the paper is twofold: a priori error estimate for the spectral approximation is derived; a conjugate gradient optimization algorithm is designed to efficiently solve the discrete optimization problem. In addition, some numerical experiments are carried out to confirm the efficiency of the proposed method. The obtained numerical results show that the convergence is exponential for smooth exact solutions.
Keywords
1 Introduction
Optimal control problems (OCPs) can be found in many scientific and engineering applications, and it has become a very active and successful research area in recent years. Considerable work has been done in the area of OCPs governed by integral order differential equations, the literature on this field is huge, and it is impossible to give even a very brief review here. Recently, fractional differential equations (FDEs) have gained considerable importance due to their application in various sciences, such as control theory [1, 2], viscoelastic materials [3, 4], anomalous diffusion [5–7], advection and dispersion of solutes in porous or fractured media [8], etc. [9–12]. Therefore, the optimal control problem for the fractional-order system initiated a new research direction and has received increasing attention.
A general formulation and a solution scheme for the fractional optimal control problem (FOCP) were first proposed in [13], where the fractional variational principle and the Lagrange multiplier technique were used. Following this idea, Frederico and Torres [14, 15] formulated a Noether-type theorem in the general context and studied fractional conservation laws. Mophou [16] applied the classical control theory to a fractional diffusion equation, involving a Riemann-Liouville fractional time derivative. Dorville et al. [17] later extended the results of [16] to a boundary fractional optimal control. Also we refer the interested reader in FOCP to [18–24] for some recent work on the subject.
Recently, considerable efforts have been made in developing spectral methods for solving FDEs. For instance, a Legendre spectral approximation was proposed in [25–27] to solve the fractional diffusion equations. Bhrawy et al. [28] applied the shifted Legendre spectral collocation method to obtain the numerical solution of the space-time fractional Burger’s equation. A spectral collocation scheme based upon the generalized Laguerre polynomials was investigated in [29] to obtain a numerical solution of the fractional pantograph equation with variable coefficients on a semi-infinite domain. With the help of operational matrices of fractional derivatives for orthogonal polynomials, the Jacobi tau spectral method is also utilized in [30] to solve multi-term space-time fractional partial differential equations. On the other hand, there exist also limited but very promising efforts in developing spectral methods for solving FOCPs. In [31], a numerical direct method based on the Legendre orthonormal basis and operational matrix of Riemann-Liouville fractional integration were introduced to solve a general class of FOCP, and the convergence of the proposed method was also extensively discussed. Ye and Xu [32] proposed a Galerkin spectral method to solve the linear-quadratic FOCP associated to the time fractional diffusion equation with control constraints, and detailed error analysis was carried out. Doha et al. [33] derived an efficient numerical scheme based on the shifted orthonormal Jacobi polynomials to solve a general form of the FOCPs.
The main aim of this work is to derive a priori error estimates for spectral approximation to an unconstrained FOCP with general convex cost functional, and propose an efficient algorithm to solve the discrete control problem. As compared to the linear-quadratic FOCP considered by Ye and Xu [32], the presence of the general cost functional here leads to many additional difficulties, one of which is that the derivation of the optimality condition.
The rest of the paper is organized as follows. In the next section we formulate the optimal control problem under consideration and derive the optimality conditions. In Section 3, the space-time spectral discretization is presented. Thereafter, the main result on the error analysis for the considered optimal control problem is given in Section 4. In this section, error estimates for the error in the control, state, and adjoint variables are analyzed. In Section 5, we describe the overall algorithm and present some numerical examples to illustrate our results. Some concluding remarks are given at the end of this article.
2 Fractional optimal control problem and optimization
Lemma 2.1
Proof
In what follows we will need the mapping \(q\rightarrow u(q) \rightarrow z(q)\), where for any given q, \(u(q)\) is defined by (2.9), and once \(u(q)\) is known \(z(q)\) is defined by (2.22).
3 Space-time spectral discretization
In this section we investigate a space-time spectral approximation to the optimization problem (2.12).
4 A priori error estimation
We now carry out an error analysis for the spectral approximation (3.3). To simplify the notations, we let c be a generic positive constant independent of any functions and of any discretization parameters. We use the expression \(A\lesssim B\) to mean that \(A\leq cB\).
Following [25], the error between the solution of (2.9) and the solution of (4.1) can be estimated as follows.
Lemma 4.1
We are now in a position to analyze the approximation error of the proposed space-time spectral method. The proof of the main result will be accomplished with a series of lemmas which we present below.
Lemma 4.2
Proof
Lemma 4.3
Proof
Lemma 4.4
Proof
The proof goes along the same lines as Lemma 4.5 in [35]. □
Using the above lemmas and following the same lines as the proof of Theorem 4.1 in [35], we obtain the main result concerning the approximation errors.
Theorem 4.1
5 Conjugate gradient optimization algorithm and numerical results
5.1 Conjugate gradient optimization algorithm
In the following, we propose a conjugate gradient algorithm for the associated linear-quadratic optimization problem. The details are described below.
The overall process is summarized below.
Conjugate gradient optimization algorithm
- (a)
Solve problems (5.6)-(5.7), let \(d_{L}^{(k)}=z_{L}(q_{L}^{(k)})+ q_{L}^{(k)}\), \(s_{L}^{(k)}=g_{L}^{(k)}\).
- (b)
Solve problems (5.4)-(5.5), and set \(\tilde{d}_{L}^{(k)}=\tilde{z}_{L}^{(k)}+ s_{L}^{(k)}\), \(\rho_{k}=\frac{ (d_{L}^{(k)},d_{L}^{(k)} )_{\Omega}}{ (\tilde{d}_{L}^{(k)},s_{L}^{(k)} )_{\Omega}}\).
- (c)
Update \(q_{L}^{(k+1)}=q_{L}^{(k)}-\rho_{k} s_{L}^{(k)}\), \(d_{L}^{(k+1)}=d_{L}^{(k)}-\rho_{k} \tilde{d}_{L}^{(k)}\).
- (d)
If \(\Vert d_{L}^{(k+1)}\Vert _{0,\Omega}\leq\) tolerance, then take \(q_{L}^{\ast}=q_{L}^{(k+1)}\), and solve problems (3.2) and (3.6) to get \(u_{L}(q_{L}^{\ast})\) and \(z_{L}(q_{L}^{\ast})\); else, let \(\beta_{k}=\frac{\Vert d_{L}^{(k+1)}\Vert ^{2}_{0,\Omega}}{ \Vert d_{L}^{(k)}\Vert ^{2}_{0,\Omega}}\), \(s_{L}^{(k+1)}=d_{L}^{(k+1)}+\beta_{k} s_{L}^{(k)}\). Set \(k=k+1\), repeat (a)-(d).
5.2 Numerical results
We are now in a position to carry out some numerical experiments and present some results to validate the obtained error estimates. In all the calculations, we take \(T=1\).
Example 5.1
For this choice of data, that is, the exact solution \(u(q^{\ast})\) serves as the observation data, problem (2.1)-(2.2) is indeed an inverse problem about unknown parameter in the right-hand side and the corresponding objective function is expected to attain its minimum 0.
In this example, we fix the initial guess at \(q^{(0)}=1\). We should mention here that any initial guess is possible. We will show in our next example that the presence of perturbation or noise in the control has limited influences on the optimization algorithm.
Maximum absolute errors for q , u , and J at \(\pmb{N=20}\) , \(\pmb{\alpha =0.3}\) , and various choices of M for Example 5.1
M | 4 | 6 | 8 | 10 | 12 | 14 | 16 |
---|---|---|---|---|---|---|---|
q | 4.08E − 03 | 5.24E − 05 | 2.93E − 07 | 1.58E − 09 | 9.97E − 12 | 5.17E − 14 | 2.30E − 14 |
u | 4.75E − 02 | 7.96E − 04 | 1.17E − 05 | 1.39E − 07 | 1.28E − 09 | 9.82E − 12 | 3.43E − 13 |
J | 2.86E − 05 | 5.77E − 09 | 8.23E − 13 | 8.83E − 17 | 6.38E − 21 | 3.09E − 25 | 1.12E − 29 |
Maximum absolute errors for q , u , and J at \(\pmb{M=20}\) and various choices of α and N for Example 5.1
Errors | α | N = 4 | N = 6 | N = 8 | N = 10 | N = 12 | N = 14 |
---|---|---|---|---|---|---|---|
q | 0.1 | 7.11E − 06 | 8.06E − 08 | 6.19E − 10 | 3.27E − 12 | 1.79E − 14 | 7.75E − 15 |
0.5 | 1.68E − 05 | 2.14E − 07 | 1.61E − 09 | 8.24E − 12 | 2.94E − 14 | 4.91E − 16 | |
0.99 | 6.96E − 05 | 1.07E − 06 | 8.98E − 09 | 4.70E − 11 | 1.63E − 13 | 2.89E − 16 | |
u | 0.1 | 7.74E − 05 | 9.14E − 07 | 7.45E − 09 | 3.98E − 11 | 2.14E − 13 | 8.67E − 14 |
0.5 | 1.82E − 04 | 2.53E − 06 | 1.97E − 08 | 1.00E − 10 | 3.66E − 13 | 1.14E − 14 | |
0.99 | 6.69E − 04 | 9.16E − 06 | 7.07E − 08 | 3.82E − 10 | 1.75E − 12 | 5.11E − 15 | |
J | 0.1 | 8.39E − 11 | 8.02E − 15 | 3.24E − 19 | 6.35E − 24 | 6.76E − 29 | 1.98E − 31 |
0.5 | 4.64E − 10 | 6.15E − 13 | 3.01E − 18 | 6.74E − 23 | 7.83E − 28 | 3.16E − 32 | |
0.99 | 5.00E − 09 | 7.90E − 13 | 3.88E − 17 | 8.27E − 22 | 9.04E − 27 | 6.78E − 32 |
Example 5.2
Maximum absolute errors for q , u , and J at \(\pmb{N=20}\) , \(\pmb{\alpha =0.6}\) and various choices of M for Example 5.2
M | 4 | 6 | 8 | 10 | 12 | 14 | 16 |
---|---|---|---|---|---|---|---|
q | 7.24E − 02 | 1.18E − 03 | 1.62E − 05 | 1.92E − 07 | 1.75E − 09 | 1.34E − 11 | 1.57E − 13 |
u | 1.30E − 01 | 2.17E − 03 | 3.19E − 05 | 3.78E − 07 | 3.48E − 09 | 2.66E − 11 | 2.22E − 13 |
J | 9.47E − 02 | 1.07E − 03 | 2.72E − 06 | 2.30E − 09 | 1.08E − 12 | 6.66E − 16 | 8.88E − 16 |
Maximum absolute errors for q , u , and J at \(\pmb{M=20}\) and various choices of α and N for Example 5.2
Errors | α | N = 4 | N = 6 | N = 8 | N = 10 | N = 12 |
---|---|---|---|---|---|---|
q | 0.2 | 9.18E − 02 | 1.01E − 05 | 5.45E − 08 | 1.76E − 10 | 3.72E − 13 |
0.6 | 2.06E − 03 | 2.55E − 05 | 1.50E − 07 | 5.13E − 10 | 1.15E − 12 | |
0.9 | 3.50E − 03 | 4.68E − 05 | 2.85E − 07 | 9.94E − 10 | 2.23E − 12 | |
u | 0.2 | 8.55E − 05 | 8.52E − 07 | 4.46E − 09 | 1.40E − 11 | 9.37E − 14 |
0.6 | 1.32E − 04 | 1.23E − 06 | 5.65E − 09 | 1.55E − 11 | 2.35E − 14 | |
0.9 | 1.01E − 04 | 1.24E − 06 | 7.83E − 09 | 2.66E − 11 | 6.33E − 14 | |
J | 0.2 | 3.17E − 05 | 2.37E − 08 | 3.38E − 11 | 4.55E − 14 | 1.33E − 15 |
0.6 | 5.52E − 05 | 3.85E − 08 | 2.22E − 11 | 2.08E − 13 | 2.44E − 15 | |
0.9 | 8.53E − 05 | 8.61E − 08 | 2.57E − 11 | 9.77E − 14 | 2.22E − 16 |
6 Concluding remarks
In the present work, we have shown an efficient optimization algorithm for the space-time fractional equation optimal control problem based on the spectral approximation. A priori error estimates are derived. Some numerical experiments have been carried out to confirm the theoretical results.
There are many important issues that still need to be addressed. Firstly, the same formulation and solution scheme can be used with minor changes for the problem defined in terms of Riemann-Liouville derivatives. Secondly, studies for more complicated control problems and constraint sets are needed. Thirdly, although our analysis and algorithm are designed for the optimization of the distributed control problem, we hope that they are generalizable for the minimization problems of other parameters, such as boundary conditions and so on.
Declarations
Acknowledgements
The work of Xingyang Ye is partially supported by the Science Foundation of Jimei University, China (Grant Nos. ZQ2013005 and ZC2013021), Foundation (Class B) of Fujian Educational Committee (Grant No. FB2013005), Foundation of Fujian Educational Committee (No. JA14180). The work of Chuanju Xu was partially supported by National NSF of China (Grant Nos. 11471274 and 11421110001).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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