Departmentof Mathematics

Introduction

The problem at hand is an optimal control problem in which the state is determined by variational inequality, viz. the elliptic obstacle problem, rather than by a partial differential equation. In fact, the variational inequality is formulated equivalently as an elliptic equation plus a complementarity system. Consequently, the optimal control problem is a function space MPCC (mathematical program with equilibrium constraints).

The problem and its solution are taken from [Meyer and Thoma, 2013, Example 7.1].

Variables & Notation

Unknowns

$$\begin{array}{ccc}\hfill u\in L^2\left(\Omega \right)& \text{control variable}\hfill & \hfill \\ \hfill y\in H_0^1\left(\Omega \right)& \text{state variable}\hfill \\ \hfill \xi \in L^2\left(\Omega \right)& \text{slack variable}\hfill \end{array}$$

Given Data

The given data is chosen in a way which admits an analytic solution.

$$\begin{array}{ccccc}\hfill \Omega & ={\left(0,1\right)}^2\hfill & \hfill & \text{computational domain}\hfill & \hfill \\ \hfill \Gamma & \hfill & \hfill & \text{its boundary}\hfill \\ \hfill {\Omega }_1& \text{(see below)}\hfill & \hfill & \text{subdomain of }\Omega \hfill \\ \hfill {\Omega }_2& =\left(0.0,0.5\right)\times \left(0.0,0.8\right)\hfill & \hfill & \text{subdomain of }\Omega \hfill \\ \hfill {\Omega }_3& =\left(0.5,1.0\right)\times \left(0.0,0.8\right)\hfill & \hfill & \text{subdomain of }\Omega \hfill \\ \hfill \end{array}$$

$$\begin{array}{ccccc}\hfill y_d\left(x\right)& =\left\{\begin{array}{cc}-400\left(\right.q_1\left(y_1\right)+q_2\left(y_2\right)\left)\right.{\left|\right.}_{y=Q^{\top <!--FUNCTION\; APPLICATION-->}\left(x-\hat{x}\right)+\hat{x}},\hfill & x\in {\Omega }_1\hfill \\ z_1\left(x_1\right)z_2\left(x_2\right),\hfill & x\in {\Omega }_2\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}\right.\hfill & \hfill & \text{desired state (discontinuous)}\hfill & \hfill \\ \hfill u_d\left(x\right)& =\left\{\begin{array}{cc}{p}_1\left(Q^{\top <!--FUNCTION\; APPLICATION-->}\left(x-\hat{x}\right)+\hat{x}\right),\hfill & x\in {\Omega }_1\hfill \\ -z_1^{″<!--FUNCTION\; APPLICATION-->}\left(x_1\right)-z_2^{″<!--FUNCTION\; APPLICATION-->}\left(x_2\right),\hfill & x\in {\Omega }_2\hfill \\ -z_1\left(x_1-0.5\right)z_2\left(x_2\right),\hfill & x\in {\Omega }_3\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}\right.\hfill & \hfill & \text{desired control (discontinuous)}\hfill \\ \hfill \end{array}$$

The subdomain ${\Omega }_1$ is a square with midpoint $\hat{x}=\left(0.8,0.9\right)$ and edge length 0.1, which has been rotated about its midpoint by 30 degrees in counter-clockwise direction. The four vertices of ${\Omega }_1$ can thus be obtained from

$$\left(\begin{array}{cccc}\hfill \hat{x}\hfill & \hfill \hat{x}\hfill & \hfill \hat{x}\hfill & \hfill \hat{x}\hfill \end{array}\right)+Q\left(\begin{array}{cccc}\hfill -0.05& \hfill 0.05& \hfill 0.05& \hfill -0.05\\ \hfill -0.05& \hfill -0.05& \hfill 0.05& \hfill 0.05\end{array}\right)\approx \left(\begin{array}{cccc}\hfill 0.7817& \hfill 0.8683& \hfill 0.8183& \hfill 0.7317\\ \hfill 0.8317& \hfill 0.8817& \hfill 0.9683& \hfill 0.9183\end{array}\right)$$

with the rotation matrix

$$Q=\left(\begin{array}{cc}\hfill \cos <!--FUNCTION\; APPLICATION-->\frac{\pi }{6}& \hfill -\sin <!--FUNCTION\; APPLICATION-->\frac{\pi }{6}\\ \hfill \sin <!--FUNCTION\; APPLICATION-->\frac{\pi }{6}& \hfill \cos <!--FUNCTION\; APPLICATION-->\frac{\pi }{6}\end{array}\right).$$

Note that ${\Omega }_1$ does not intersect ${\Omega }_2$ nor ${\Omega }_3$. The remaining pieces of data are

$$\begin{array}{ccc}\hfill z_1\left(x_1\right)& =-4096x_1^6+6144x_1^5-3072x_1^4+512x_1^3\hfill & \hfill \\ \hfill z_2\left(x_2\right)& =-244.140625x_2^6+585.937500x_2^5-468.750x_2^4+125x_2^3\hfill \\ \hfill q_1\left(y_1\right)& =-200{\left(y_1-0.8\right)}^2+0.5\hfill \\ \hfill q_2\left(y_2\right)& =-200{\left(y_2-0.9\right)}^2+0.5\hfill \\ \hfill p_1\left(y_1,y_2\right)& =q_1\left(y_1\right)q_2\left(y_2\right).\hfill \end{array}$$

Problem Description

$$\begin{array}{ccc}\hfill \text{Minimize}& \frac{1}{2}\parallel y-y_d{\parallel }_{L^2\left(\Omega \right)}^2+\frac{1}{2}\parallel u-u_d{\parallel }_{L^2\left(\Omega \right)}^2\hfill & \hfill \\ \hfill \text{s.t.}& \left\{\begin{array}{ccccc}\hfill -△y& =u+\xi \hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill y& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill y\ge 0,\xi & \ge 0,y\xi =0\hfill & \hfill & \text{in }\Omega .\hfill \end{array}\right.\hfill \end{array}$$

Optimality System

Besides the state $y\in H_0^1\left(\Omega \right)$, control $u\in L^2\left(\Omega \right)$ and slack variable $\xi \in L^2\left(\Omega \right)$, the optimality system consists of the adjoint state $p\in H_0^1\left(\Omega \right)$ and a Lagrange multiplier $\mu \in H^{-1}\left(\Omega \right)$ pertaining to the constraint $y\ge 0$. The adjoint state $p$ serves a double role, since it also acts as Lagrange multiplier for the pointwise constraint $\xi \ge 0$. As usual for MPCCs, no multiplier is introduced for the constraint $y\xi =0$.

It should be noted that for MPCCs, a canocical first-order optimality condition does not exist. The following system represents a particular set of first-order necessary conditions, viz. of strongly stationary type.

$$\begin{array}{ccccc}\hfill -△y& =u+\xi \hfill & \hfill & \text{in }\Omega \hfill & \hfill \\ \hfill y& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill -△p& =y-y_d+\mu \hfill & \hfill & \text{in }\Omega \hfill \\ \hfill p& =0\hfill & \hfill & \text{on }\partial \Omega \hfill \\ \hfill u-u_d-p& =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill y\ge 0,\xi \ge 0,y\xi & =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill \mu y& =0\hfill & \hfill & \text{in }\Omega \text{ a weak sense}\hfill \\ \hfill p\xi & =0\hfill & \hfill & \text{in }\Omega \hfill \\ \hfill p& \ge 0\hfill & \hfill & \text{in }B\hfill \\ \hfill \mu & \le 0\hfill & \hfill & \text{in }B\text{ in a weak sense}.\hfill \end{array}$$

The set $B=\left\{x\in \Omega :y\left(x\right)=\xi \left(x\right)=0\right\}$ is termed the bi-active set. It is the last two conditions on the signs of $p$ and $\mu$ which are particular for the concept of strong stationarity.

Since $\mu$ belongs only to $H^{-1}\left(\Omega \right)$, two of the conditions above must be imposed in a weak sense. This can be done in the following way:

$$\begin{array}{ccc}\hfill {⟨\mu ,v⟩}_{H^{-1}\left(\Omega \right),H_0^1\left(\Omega \right)}& =0\text{for all }v\in H_0^1\left(\Omega \right)\text{ satisfying }v\left(x\right)=0\text{ where }y\left(x\right)=0\hfill & \hfill \\ \hfill {⟨\mu ,v⟩}_{H^{-1}\left(\Omega \right),H_0^1\left(\Omega \right)}& \le 0\text{for all }v\in H_0^1\left(\Omega \right)\text{ satisfying }v\left(x\right)\ge 0\text{ where }y\left(x\right)=0\hfill \\ \hfill & \text{and }v\left(x\right)=0\text{ where }\xi \left(x\right)>0.\hfill \end{array}$$

Supplementary Material

The following functions given in [Meyer and Thoma, 2013, Example 7.1] satisfy the set of necessary optimality conditions of strongly stationary type above. An important feature of this selection is that there is a nontrivial bi-active set:

$$B=\left\{x\in \Omega :y\left(x\right)=\xi \left(x\right)=0\right\}=\left(0.0,1.0\right)\times \left(0.8,1.0\right).$$

Moreover, second-order optimality conditions have been verified, and thus $\left(y,\xi ,u\right)$ is guaranteed to represent a local minimum.

$$\begin{array}{ccccc}\hfill y& =\left\{\begin{array}{cc}{z}_1\left(x_1\right)z_2\left(x_2\right),\hfill & x\in {\Omega }_2\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}\right.\hfill & \hfill & \text{(of class }{C}^2\left(\overline{\Omega }\right)\text{)}\hfill & \hfill \\ \hfill u& =\left\{\begin{array}{cc}-z_1^{″<!--FUNCTION\; APPLICATION-->}\left(x_1\right)-z_2^{″<!--FUNCTION\; APPLICATION-->}\left(x_2\right),\hfill & x\in {\Omega }_2\hfill \\ -z_1\left(x_1-0.5\right)z_2\left(x_2\right),\hfill & x\in {\Omega }_3\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}\right.\hfill & \hfill & \hfill \\ \hfill \xi & =\left\{\begin{array}{cc}{z}_1\left(x_1-0.5\right)z_2\left(x_2\right),\hfill & x\in {\Omega }_3\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}\right.\hfill & \hfill & \text{(continuous)}\hfill \\ \hfill p& =\left\{\begin{array}{cc}{p}_1\left(Q^{\top <!--FUNCTION\; APPLICATION-->}x\right),\hfill & x\in {\Omega }_1\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}\right.\hfill & \hfill & \text{(continuous, but not }{C}^1\left(\Omega \right)\text{)}\hfill \\ \hfill {⟨\mu ,v⟩}_{H^{-1}\left(\Omega \right),H_0^1\left(\Omega \right)}& ={\int \; <!--nolimits-->}_{\partial {\Omega }_1}\nabla <!--FUNCTION\; APPLICATION-->p{|}_{{\Omega }_1}\cdot n_{1}v\mathrm{\text{<mstyle class="text"><mtext class="textup" mathvariant="normal" >d<mi>s</mi><mo class="MathClass-punc">,</mo> </mtext class="textup" mathvariant="normal" ></mstyle >}}\hfill \end{array}$$

where $n_1$ is the unit outer normal to the rotated square subdomain ${\Omega }_1$. Note that $\mu$ is a line functional concentrated on $\partial {\Omega }_1$. In more explicit terms, it can be expressed as

The remaining data are

$$\begin{array}{ccc}\hfill z_1^{″<!--FUNCTION\; APPLICATION-->}\left(x_1\right)& =-122880x_1^4+122880x_1^3-36864x_1^2+3072x_1\hfill & \hfill \\ \hfill z_2^{″<!--FUNCTION\; APPLICATION-->}\left(x_2\right)& =-7324.218750x_2^4+11718.75x_2^3-5625x_2^2+750x_2^1\hfill \\ \hfill q_1^{\prime }\left(x_1\right)& =-400\left(x_1-0.8\right)\hfill \\ \hfill q_2^{\prime }\left(x_2\right)& =-400\left(x_2-0.9\right).\hfill \end{array}$$

Revision History

• 2021–02–11: fixed typo in transformation of data $y_d$ and $u_d$ on ${\Omega }_1$
• 2013–03–01: problem added to the collection

References