Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

Radouen Ghanem; Billel Zireg

doi:10.3934/naco.2020002

Article Contents

2020, Volume 10, Issue 3: 275-300. Doi: 10.3934/naco.2020002

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Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

Radouen Ghanem and
Billel Zireg

Numerical Analysis, Optimization and Statistical Laboratory (LANOS), Badji-Mokhtar, Annaba University, P.O. Box 12, 23000, Annaba, Algeria

Received: December 2018

Revised: August 2019

Published: June 2020

The authors would like to thank the anonymous referee for careful reading and the suggestions of some improvements in presentation that have been implemented in the final version of the manuscript

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Abstract

This work is deals with the numerical solution of a bilateral obstacle optimal control problem which is similar to the one given in Bergounioux et al [9] with some modifications. It can be regarded as an extension of our previous work [18], where the main feature of the present work is that the controls and the two obstacles are the same. For the numerical resolution we follow the idea of our previous work [18]. We begin by discretizing the optimality system of the underlying problem by using finite differences schemes, then we propose an iterative algorithm. Finally, numerical examples are provides to show the efficiency of the proposed algorithm and the used scheme.

Keywords:

Mathematics Subject Classification: Primary: 65K15 49K10, 35J87; Secondary: 49J20, 65L12, 49M15.

Citation:

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Figure 1. Left (error $ \epsilon _{n} $, continuous line; $ \omega = 0.25 $, dash line; $ \omega = 0.5 $, dash-dot line; $ \omega = 0.75 $), right (state function $ y $)

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Figure 2. Left (obstacle function $ \psi $), right (obstacle function $ \varphi $)

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Figure 3. Left (error $ \epsilon _{n} $, continuous line; $ N = 30 $, dash line; $ N = 35 $, dash-dot line; $ N = 40 $), right (state function $ y $)

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Figure 4. Left (obstacle function $ \psi $), right (obstacle function $ \varphi $)

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Figure 5. Left(error $ \epsilon _{n} $, continuous line $ \nu = 0.1 $; dash line $ \nu = 0.5 $; dash-dot line $ \nu = 1 $), right (state function $ y $)

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Figure 6. Left(obstacle function $ \varphi $), right (obstacle function $ \psi $)

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Figure 7. Left(error $ \epsilon_{n} $, continuous line; $ \delta = h^{2} $, dash line; $ \delta = h^{3} $, dash-dot line; $ \delta = h^{4} $), right (state function $ y $)

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Figure 8. Left (obstacle function $ \psi $), right (right(obstacle function $ \varphi $))

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Algorithm 2 Implemented continuous algorithm

1: Begin
2: Input :$ \left\{ y_{0}^{\delta }, p_{0}^{\delta }, \varphi _{0}^{\delta }, \lambda _{0}^{\delta },\psi _{0}^{\delta },\delta ,\nu ,\omega_{y},\omega_{\varphi},\omega_{\psi},\varepsilon\right\} $ choose $ \varphi _{0}^{\delta } $ and $ \psi _{0}^{\delta }\in \mathcal{W},\varepsilon $ and $ \delta $ in $ \mathbb{R}_{+}^{\ast }; $
3: Calculate

$ J_{n-1} \leftarrow J_{n-1}\left(y^{\delta}_{n-1}, \varphi^{\delta}_{n-1}, \psi^{\delta}_{n-1}\right) $

4: Solve

$ \left( A+\beta _{\delta }^{\prime }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left(\psi _{n-1}^{\delta }- y_{n-1}^{\delta }\right) \right)r_{n}^{\delta }= $

$ -\omega_{y}\left( A y_{n-1}^{\delta }+\beta _{\delta }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta } \right)-\beta _{\delta }\left( \psi _{n-1}^{\delta } -y_{n-1}^{\delta }\right) -f \right) $ on $ r_{n}^{\delta } $.
5: Calculate

$ y_{n}^{\delta }=y_{n-1}^{\delta }+ $

$ r_{n}^{\delta } $.
6: Solve $ \left( A +\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta } -y_{n}^{\delta }\right) \right) p_{n}^{\delta }=y_{n}^{\delta }-z $ on $ p_{n}^{\delta } $.
7: Calculate $ \lambda_{n}^{\delta } = \nu \Delta \varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta } $.
8: Solve$ \left( \nu \Delta+\beta _{\delta }^{\prime \prime }\left(\psi _{n-1}^{\delta } -y_{n}^{\delta }\right) p_{n}^{\delta }\right)r_{n}^{\delta }=-\omega_{\psi}\left(\nu A_{h}^{d}\psi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta }-y_{n}^{\delta }\right) p_{n}^{\delta }+\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
9: Calculate

$ \psi _{n}^{\delta }=\psi _{n-1}^{\delta }+ $

$ r_{n}^{\delta } $.
10: Solve$ \left( \nu \Delta-\beta _{\delta }^{\prime \prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta } \right) p_{n}^{\delta }\right) r_{n}^{\delta }=-\omega_{\varphi}\left(\nu A_{h}^{d}\varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta }-\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
11: Calculate

$ \varphi _{n}^{\delta }=\varphi _{n-1}^{\delta }+ $

$ r_{n}^{\delta } $.
12: Calculate $ J_{n} \leftarrow J_{n-1}\left(y^{\delta}_{n}, \varphi^{\delta}_{n},\psi^{\delta}_{n}\right) $.
13: If

$ |J_{n}-J_{n-1}| \leq \varepsilon $

Stop.
14: Ensure :

$ s_{n}^{\delta }=\left( y_{n}^{\delta },\varphi_{n}^{\delta },\psi_{n}^{\delta },p_{n}^{\delta }\right) $

is a solution.
15: Else; $ n\leftarrow n+1 $, go to Begin.
16: End if
17: End

| Show Table

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Table 1. Numerical results for two dimensional space while varying $ \omega $

$ \omega $	$ \sharp $ Iteration	$ {J} $	$ \mid J_{n}-J_{n-1}\mid $	$ {\epsilon}_{n} $
0.25	62	29.460551e-4	9.601694e-16	3.982865e-11
0.5	28	29.460551e-4	6.392456e-16	1.180507e-11
0.75	15	29.460551e-4	5.212844e-16	5.253112e-12
1	5	29.460551e-4	6.878178e-16	9.642406e-10

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Table 2. Numerical results for two dimensional space while varying $ N $

$ N $	$ \sharp $ Iteration	$ {J} $	$ \mid J_{n}-J_{n-1}\mid $	$ {\epsilon}_{n} $
30	26	1.632e-3	5.193328e-16	7.355250e-12
35	27	2.241e-3	5.165139e-16	6.513361e-12
40	28	2.946e-3	6.392456e-16	1.180507e-11
45	30	3.746e-3	7.350890e-16	2.389886e-11

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Table 3. Numerical results for two dimensional space while varying $ \nu $

$ \nu $	$ \sharp $ Iteration	$ {J} $	$ \mid J_{n}-J_{n-1}\mid $	$ {\epsilon}_{n} $
0.001	300	29.460541e-4	1.227802e-13	3.695397e-05
0.01	29	29.460541e-4	5.676882e-16	5.966039e-11
0.1	29	29.460544e-4	5.676882e-16	5.966039e-11
0.5	28	29.460549e-4	7.650130e-16	2.529592e-11
1	28	29.460551e-4	6.392456e-16	1.180507e-11

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Table 4. Numerical results for two dimensional space while varying $ \delta $

$ \delta $	$ \sharp $ Iteration	$ {J} $	$ \mid J_{n}-J_{n-1}\mid $	$ {\epsilon}_{n} $
$ h^{3} $	27	29.460541e-4	8.135853e-16	8.627484e-12
$ h^{3.5} $	27	29.460543e-4	8.270294e-16	8.809409e-12
$ h^{4} $	28	29.460551e-4	6.392456e-16	1.180507e-11
$ h^{4.5} $	33	29.460566e-4	9.358833e-16	1.016003e-10

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