Numerical study of polynomial feedback laws for a bilinear control problem

Tobias Breiten; Karl Kunisch; Laurent Pfeiffer

doi:10.3934/mcrf.2018023

Article Contents

2018, Volume 8, Issue 3&4: 557-582. Doi: 10.3934/mcrf.2018023

This issue Previous Article Necessary conditions for infinite horizon optimal control problems with state constraints Next Article General boundary value problems of the Korteweg-de Vries equation on a bounded domain

Numerical study of polynomial feedback laws for a bilinear control problem

1.
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
2.
RICAM Institute, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040 Linz, Austria

* Corresponding author: Karl Kunisch
* Corresponding author: Karl Kunisch

Received: September 2017

Revised: July 2018

Published: September 2018

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Abstract

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

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Mathematics Subject Classification: Primary: 49J20, 49N35, 93B40, 93D15.

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Figure 1. 1D Fokker-Planck equation

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Figure 2. Comparison of the original and reduced models, for $n = 100$, $r = 25$, and $\beta = 10^{-4}$

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Figure 3. Comparison of the original and reduced models, for $n = 100$, $r = 25$, and $\beta = 10^{-4}$

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Figure 4. Comparison of the reduced models with $r = 21$ and $r = 9$, derived from a finer discretization (with $n = 1000$), with the setup of Figure 3

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Figure 5. Convergence of the control laws for $\beta = 10^{-4}, n = 1000$ and $r = 9.$

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Figure 6. Convergence of the control laws for $\beta = 10^{-4}, n = 1000$ and $r = 9.$

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Figure 7. Initial condition and controls for the test case 2

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Figure 8. Initial condition and controls for the test case 3

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Figure 9. 2D Fokker-Planck equation

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Figure 10. Control shape functions α₁ and α₂

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Figure 11. Singular value decay for n = 2500

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Figure 12. Initial condition and controls for the test case 4

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Figure 13. Initial condition and controls for the test case 5

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Table 1. Convergence results for the test case 1

$\beta$	$J(u_2)$	$J(u_3)$	$J(u_4)$	$J(u_5)$	$J(u_6)$	$J(u_{\text{opt}})$
1e$^{-3}$	0.038	0.038	0.038	0.038	0.038	0.038
1e$^{-4}$	0.034	0.033	0.033	0.033	0.033	0.032
1e$^{-5}$	0.037	0.031	0.031	0.031	0.031	0.030
(A) Cost of the controls $u_p$.
$\beta$		$\\| u_p-u_{\text{opt}} \\|_{L^2(0, T)}$
$\beta$		$p=2$	$p=3$	$p=4$	$p=5$	$p=6$
1e$^{-3}$		0.228	0.026	0.024	0.024	0.024
1e$^{-4}$		4.26	1.19	0.82	0.61	0.61
1e$^{-5}$		29.8	10.3	7.91	4.70	4.05
(B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.

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Table 2. Convergence results for the test case 2

$\beta$	$J(u_2)$	$J(u_3)$	$J(u_4)$	$J(u_5)$	$J(u_6)$	$J(u_{\text{opt}})$
1e$^{-3}$	0.156	0.155	0.155	0.155	0.155	0.154
5e$^{-4}$	0.147	0.145	0.145	0.145	0.145	0.144
1e$^{-4}$	0.138	0.122	0.120	0.120	0.120	0.119
5e$^{-5}$	0.190	0.114	0.111	0.112	0.111	0.110
1e$^{-5}$	0.205	0.194	0.104	0.111	0.113	0.095
(A) Cost of the controls u_p.
$\beta$		$\\| u_p-u_{\text{opt}} \\|_{L^2(0, T)}$
$\beta$		$p=2$	$p=3$	$p=4$	$p=5$	$p=6$
1e$^{-3}$		1.149	0.169	0.119	0.034	0.031
5e$^{-4}$		2.583	0.737	0.171	0.336	0.219
1e$^{-4}$		18.50	7.02	3.16	4.01	1.52
5e$^{-5}$		46.87	13.18	8.40	8.17	2.65
1e$^{-5}$		90.5	78.0	39.0	42.6	34.3
(B) L²-distance between the controls u_p and the optimal control u_opt.

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Table 3. Convergence results for the test case 3

$\beta$	$J(u_2)$	$J(u_3)$	$J(u_4)$	$J(u_5)$	$J(u_6)$	$J(u_{\text{opt}})$
1e$^{-3}$	0.525	0.511	0.511	0.512	0.510	0.507
5e$^{-4}$	0.451	0.417	0.431	0.459	0.446	0.408
1e$^{-4}$	0.381	0.368	2.689	$\infty$	$\infty$	0.246
5e$^{-5}$	0.381	0.432	$\infty$	$\infty$	$\infty$	0.206
1e$^{-5}$	0.365	$\infty$	$\infty$	$\infty$	$\infty$	0.147
(A) Cost of the controls $u_p$.
$\beta$		$\\| u_p-u_{\text{opt}} \\|_{L^2(0, T)}$
$\beta$		$p=2$	$p=3$	$p=4$	$p=5$	$p=6$
1e$^{-3}$		4.88	1.50	1.77	2.31	1.52
5e$^{-4}$		11.26	5.03	7.11	11.89	11.99
1e$^{-4}$		46.34	35.36	57.08	$\infty$	$\infty$
5e$^{-5}$		74.79	60.86	$\infty$	$\infty$	$\infty$
1e$^{-5}$		172.3	$\infty$	$\infty$	$\infty$	$\infty$
(B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.

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Table 4. Convergence results for the test case 4

$\beta$	$J(u_2)$	$J(u_3)$	$J(u_4)$	$J(u_5)$	$J(u_{\text{opt}})$
1e$^{-3}$	0.247	0.235	0.234	0.234	0.232
5e$^{-4}$	0.232	0.207	0.205	0.205	0.203
1e$^{-4}$	0.252	0.180	0.174	0.174	0.171
5e$^{-5}$	0.279	0.179	0.168	0.168	0.165
1e$^{-5}$	0.524	0.182	20.696	0.164	0.158
(A) Cost of the controls $u_p$.
$\beta$		$\\| u_p-u_{\text{opt}} \\|_{L^2(0, T)}$
$\beta$		$p=2$	$p=3$	$p=4$	$p=5$
1e$^{-3}$		3.53	0.80	0.19	0.14
5e$^{-4}$		6.73	1.42	0.37	0.24
1e$^{-4}$		27.40	5.78	1.83	1.24
5e$^{-5}$		52.50	11.06	3.69	2.40
1e$^{-5}$		257.01	63.97	84.31	10.61
(B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.

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Table 5. Convergence results for the test case 5

$\beta$	$J(u_2)$	$J(u_3)$	$J(u_4)$	$J(u_5)$	$J(u_{\text{opt}})$
1e$^{-1}$	7.58	7.57	7.57	7.57	7.52
5e$^{-2}$	6.41	6.39	6.40	6.39	6.35
1e$^{-2}$	3.70	3.34	3.09	3.32	3.00
5e$^{-3}$	3.07	2.68	2.28	2.96	2.05
1e$^{-3}$	2.45	2.41	$\infty$	$\infty$	0.93
(A) Cost of the controls $u_p$.
$\beta$		$\\| u_p-u_{\text{opt}} \\|_{L^2(0, T)}$
$\beta$		$p=2$	$p=3$	$p=4$	$p=5$
1e$^{-1}$		0.70	0.61	0.62	0.60
5e$^{-2}$		1.10	0.69	0.80	0.63
1e$^{-2}$		13.02	11.10	4.08	9.01
5e$^{-3}$		21.59	19.80	9.66	20.06
1e$^{-3}$		47.34	55.69	$\infty$	$\infty$
(B) $L^2$-distance between the controls $u_p$ and the optimal control $u_{\text{opt}}$.

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