Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

Zhen-Zhen Tao; Bing Sun

doi:10.3934/dcdsb.2022080

Article Contents

2023, Volume 28, Issue 1: 359-384. Doi: 10.3934/dcdsb.2022080

This issue Previous Article New well-posedness results for stochastic delay Rayleigh-Stokes equations Next Article Upper semi-continuity of non-autonomous fractional stochastic

$ p $

-Laplacian equation driven by additive noise on

$ \mathbb{R}^n $

Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

Zhen-Zhen Tao^1, and
Bing Sun^2,3, ,

1.
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China
2.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China
3.
MIIT Key Laboratory of MTCIS, Beijing Institute of Technology, Beijing 102488, China

^* Corresponding author: Bing Sun
^* Corresponding author: Bing Sun

Received: October 31, 2021

Revised: February 28, 2022

Published: January 2023

The second author is supported in part by the National Natural Science Foundation of China under Grant No. 11471036

Abstract Full Text(HTML) Figure(4) / Table(8) Related Papers Cited by

Abstract

In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.

Keywords:

Mathematics Subject Classification: 49M25, 49M41, 65M60, 65N35.

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Figure 1. Error estimates in Example 4.1

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Figure 2. Error estimates versus $ M $ with $ N = 16 $ in Example 4.2

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Figure 3. Error estimates in Example 4.3

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Figure 4. Error estimates in Example 4.4

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Table 1. The numerical results of error estimates versus $ M $ with $ N = 16 $ in Example 4.1

$ M $	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	2.6735e-1	2.0432e-2	8.0550e-4	1.9906e-5	4.0128e-7
$ \left\\|y-y_S \right\\|_X $	2.5661e-1	1.9019e-2	7.4403e-4	1.8202e-5	3.6027e-7
$ \left\\|p-p_S \right\\|_X $	2.7749e-1	2.0434e-2	8.0541e-4	1.9922e-5	4.0364e-7

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Table 2. The numerical results of error estimates versus $ N $ with $ M = 16 $ in Example 4.1

$ N $	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	7.2398e-1	1.8638e-1	2.4561e-2	2.1601e-3	1.2982e-4
$ \left\\|y-y_S \right\\|_X $	6.6338e-1	1.6126e-1	2.1630e-2	1.8708e-3	1.1242e-4
$ \left\\|p-p_S \right\\|_X $	7.6287e-1	1.8557e-1	2.4975e-2	2.1601e-3	1.2982e-4

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Table 3. The numerical results of error estimates versus $ M $ with $ N = 16 $ in Example 4.2

$ M $	2	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	1.5461e-0	2.4357e-1	1.8796e-2	6.5755e-4	1.5708e-5	9.6694e-5
$ \left\\|y-y_S \right\\|_X $	2.3426e-1	8.7262e-3	2.6265e-3	1.0560e-3	2.4378e-4	1.6250e-4
$ \left\\|p-p_S \right\\|_X $	1.2673e-0	2.4358e-1	1.7679e-2	6.5756e-4	1.5549e-5	9.6694e-5

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Table 4. The numerical results of error estimates versus $ N $ with $ M = 16 $ in Example 4.2

$ N $	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	6.5948e-1	1.8336e-1	2.5193e-2	2.1668e-3	1.2954e-4
$ \left\\|y-y_S \right\\|_X $	1.2300e-1	5.4459e-3	1.5343e-4	4.5379e-5	4.5297e-5
$ \left\\|p-p_S \right\\|_X $	6.5948e-1	1.8336e-1	2.5193e-2	2.1667e-3	1.2953e-4

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Table 5. The numerical results of error estimates versus $ M $ with $ N = 14 $ in Example 4.3

$ M $	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	2.8277e-1	2.2323e-2	9.0317e-4	2.4526e-5	9.1509e-6
$ \left\\|y-y_S \right\\|_X $	2.2620e-1	1.7289e-2	7.0066e-4	1.8947e-5	6.8776e-6
$ \left\\|p-p_S \right\\|_X $	2.9416e-1	2.2327e-2	9.0309e-4	2.4546e-5	9.1127e-6

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Table 6. The numerical results of error estimates versus $ N $ with $ M = 14 $ in Example 4.3

$ N $	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	9.0104e-1	2.4823e-2	3.7304e-2	3.3514e-3	2.0369e-4
$ \left\\|y-y_S \right\\|_X $	6.5356e-1	1.9859e-2	2.8635e-2	2.5156e-3	1.5277e-4
$ \left\\|p-p_S \right\\|_X $	8.7113e-1	2.6477e-2	3.8175e-2	3.3540e-3	2.0369e-4

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Table 7. The numerical results of error estimates versus $ M $ with $ N = 14 $ in Example 4.4

$ M $	4	6	8	10	12
$ \left\\|u-u_S \right\\|_X $	9.2474e-1	6.1536e-2	2.4340e-3	5.5967e-5	8.9987e-7
$ \left\\|y-y_S \right\\|_X $	4.1858e-2	3.7996e-3	1.6333e-4	3.7134e-6	5.7744e-8
$ \left\\|p-p_S \right\\|_X $	8.4912e-1	6.1544e-2	2.3427e-3	5.5097e-5	8.8207e-7

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Table 8. The numerical results of error estimates versus $ N $ with $ M = 14 $ in Example 4.4

$ N $	4	6	8	10	12
$ \\|u-u_S\\|_X $	2.1487e-1	9.4348e-3	2.7567e-4	4.9387e-6	7.1424e-8
$ \\|y-y_S\\|_X $	3.1639e-1	1.3647e-2	3.7899e-4	7.1197e-6	1.0138e-7
$ \\|p-p_S\\|_X $	2.1487e-1	9.2748e-3	2.5742e-4	4.8196e-6	6.9612e-8

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