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Optimal control of a non-smooth semilinear elliptic equation

  • * Corresponding author: C. Meyer

    * Corresponding author: C. Meyer 
C. Clason was supported by the DFG under grant CL 487/2-1, and C. Christof and C. Meyer were supported by the DFG under grant ME 3281/7-1, both within the priority programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”.
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  • This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand sub-differential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

    Mathematics Subject Classification: Primary: 49K20, 49J52; Secondary: 49M15.


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  • Table 1.  Numerical results in the first example

    $h$ $\alpha$ $\gamma$ $\frac{\|y_h - y\|_{L^2}}{\|y\|_{L^2}}$ $\|p_h - p\|_{L^2}$ $\|\chi_h - \chi\|_{L^\infty, h}$ # Newton
    $3.030\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}4$ $1.152\text{e}{-}3$ $1.036\text{e}{-}5$ $8.150\text{e}{-}7$ $3$
    $1.538\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}4$ $2.962\text{e}{-}4$ $2.679\text{e}{-}6$ $8.149\text{e}{-}7$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}4$ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $8.156\text{e}{-}7$ $3$
    $3.891\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}4$ $1.893\text{e}{-}5$ $1.716\text{e}{-}7$ $8.156\text{e}{-}7$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}2 $ - - - no conv.
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}3 $ - - - no conv.
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}5 $ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $3.178\text{e}{-}6$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}6 $ $7.515\text{e}{-}5$ $6.809\text{e}{-}7$ $9.178\text{e}{-}6$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}2$ $1\text{e}{-}4$ $3.267\text{e}{-}4$ $3.241\text{e}{-}6$ $8.154\text{e}{-}7$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}3$ $1\text{e}{-}4$ $2.444\text{e}{-}4$ $2.405\text{e}{-}6$ $8.154\text{e}{-}7$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}6$ $1\text{e}{-}4$ $2.449\text{e}{-}6$ $9.204\text{e}{-}9$ $8.149\text{e}{-}7$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}8$ $1\text{e}{-}4$ $1.199\text{e}{-}7$ $9.452\text{e}{-}11$ $8.153\text{e}{-}7$ $3$
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    Table 2.  Numerical results in the second example

    $h$ $\alpha$ $\gamma$ $\frac{\|y_h - y\|_{L^2}}{\|y\|_{L^2}}$ $\frac{\|p_h - p\|_{L^2}}{\|p\|_{L^2}}$ # Newton
    $3.030\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}12$ $8.708\text{e}{-}1$ $1.606\text{e}{-}2$ $4$
    $1.538\text{e}{-}2$ $1\text{e}{-}4$ $1\text{e}{-}12$ $2.281\text{e}{-}1$ $4.541\text{e}{-}3$ $5$
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}12$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
    $3.891\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}12$ $1.469\text{e}{-}2$ $3.119\text{e}{-}4$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}6$ - - no conv.
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}8$ - - no conv.
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}10$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}4$ $1\text{e}{-}14$ $5.821\text{e}{-}2$ $1.209\text{e}{-}3$ $3$
    $7.752\text{e}{-}3$ $1\text{e}{-}2$ $1\text{e}{-}12$ $3.007\text{e}{-}3$ $1.747\text{e}{-}3$ $2$
    $7.752\text{e}{-}3$ $1\text{e}{-}3$ $1\text{e}{-}12$ $1.659\text{e}{-}2$ $1.512\text{e}{-}3$ $2$
    $7.752\text{e}{-}3$ $1\text{e}{-}5$ $1\text{e}{-}12$ $1.692\text{e}{-}1$ $8.659\text{e}{-}4$ $5$
    $7.752\text{e}{-}3$ $1\text{e}{-}6$ $1\text{e}{-}12$ - - no conv.
     | Show Table
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