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Article Contents

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

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
• 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.

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

 Citation:

• 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 $y$)

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

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

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

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

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

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 $y$)

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

 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

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

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

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

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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