## mpccdist1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Obstacle:
• nonlinear elliptic - VI operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• No time dependence.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2012-08-21 by Roland Herzog. Published on 2012-08-21

## mpccdist1 description:

### 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 Thoma2013, Example 7.1].

### Variables & Notation

#### Given Data

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

The subdomain ${\Omega }_{1}$ is a square with midpoint $\stackrel{̂}{x}=\left(0.8,0.9\right)$ 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

 $\left(\begin{array}{cccc}\hfill \stackrel{̂}{x}\hfill & \hfill \stackrel{̂}{x}\hfill & \hfill \stackrel{̂}{x}\hfill & \hfill \stackrel{̂}{x}\hfill \end{array}\right)+Q\phantom{\rule{0.3em}{0ex}}\left(\begin{array}{cccc}\hfill -0.05& \hfill 0.05& \hfill 0.05& \hfill -0.05\\ \hfill -0.05& \hfill -0.05& \hfill 0.05& \hfill 0.05\end{array}\right)\approx \left(\begin{array}{cccc}\hfill 0.7817& \hfill 0.8683& \hfill 0.8183& \hfill 0.7317\\ \hfill 0.8317& \hfill 0.8817& \hfill 0.9683& \hfill 0.9183\end{array}\right)$

with the rotation matrix

 $Q=\left(\begin{array}{cc}\hfill cos\frac{\pi }{6}& \hfill -sin\frac{\pi }{6}\\ \hfill sin\frac{\pi }{6}& \hfill cos\frac{\pi }{6}\end{array}\right).$

Note that ${\Omega }_{1}$ does not intersect ${\Omega }_{2}$ nor ${\Omega }_{3}$. The remaining pieces of data are

 $\begin{array}{ccc}\hfill {z}_{1}\left({x}_{1}\right)& =-4\phantom{\rule{1em}{0ex}}096\phantom{\rule{0.3em}{0ex}}{x}_{1}^{6}+6\phantom{\rule{1em}{0ex}}144\phantom{\rule{0.3em}{0ex}}{x}_{1}^{5}-3\phantom{\rule{1em}{0ex}}072\phantom{\rule{0.3em}{0ex}}{x}_{1}^{4}+512\phantom{\rule{0.3em}{0ex}}{x}_{1}^{3}\hfill & \hfill \\ \hfill {z}_{2}\left({x}_{2}\right)& =-244.140\phantom{\rule{1em}{0ex}}625\phantom{\rule{0.3em}{0ex}}{x}_{2}^{6}+585.937\phantom{\rule{1em}{0ex}}500\phantom{\rule{0.3em}{0ex}}{x}_{2}^{5}-468.750\phantom{\rule{0.3em}{0ex}}{x}_{2}^{4}+125\phantom{\rule{0.3em}{0ex}}{x}_{2}^{3}\hfill \\ \hfill {q}_{1}\left({y}_{1}\right)& =-200\phantom{\rule{0.3em}{0ex}}{\left({y}_{1}-0.8\right)}^{2}+0.5\hfill \\ \hfill {q}_{2}\left({y}_{2}\right)& =-200\phantom{\rule{0.3em}{0ex}}{\left({y}_{2}-0.9\right)}^{2}+0.5\hfill \\ \hfill {p}_{1}\left({y}_{1},{y}_{2}\right)& ={q}_{1}\left({y}_{1}\right)\phantom{\rule{0.3em}{0ex}}{q}_{2}\left({y}_{2}\right).\hfill \end{array}$

### 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 $y\phantom{\rule{0.3em}{0ex}}\xi =0$.

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.

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:

### Supplementary Material

The following functions given in [Meyer and Thoma2013, 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:

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

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

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

 $\begin{array}{ccc}\hfill {⟨\mu ,v⟩}_{{H}^{-1}\left(\Omega \right),{H}_{0}^{1}\left(\Omega \right)}& ={\int }_{0.75}^{0.85}Q\left(\begin{array}{c}\hfill -0.5\phantom{\rule{0.3em}{0ex}}{q}_{1}^{\prime }\left({x}_{1}\right)\hfill \\ \hfill 20\phantom{\rule{0.3em}{0ex}}{q}_{1}\left({x}_{1}\right)\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill 0\hfill \\ \hfill -1\hfill \end{array}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}v\left({x}_{1},0.85\right)\phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}{x}_{1}\hfill & \hfill \\ \hfill & \phantom{\rule{1em}{0ex}}+{\int }_{0.75}^{0.85}Q\left(\begin{array}{c}\hfill -0.5\phantom{\rule{0.3em}{0ex}}{q}_{1}^{\prime }\left({x}_{1}\right)\hfill \\ \hfill -20\phantom{\rule{0.3em}{0ex}}{q}_{1}\left({x}_{1}\right)\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}v\left({x}_{1},0.95\right)\phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}{x}_{1}\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}+{\int }_{0.85}^{0.95}Q\left(\begin{array}{c}\hfill 20\phantom{\rule{0.3em}{0ex}}{q}_{2}\left({x}_{2}\right)\hfill \\ \hfill -0.5\phantom{\rule{0.3em}{0ex}}{q}_{2}^{\prime }\left({x}_{2}\right)\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill -1\hfill \\ \hfill 0\hfill \end{array}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}v\left(0.75,{x}_{2}\right)\phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}{x}_{2}\hfill \\ \hfill & \phantom{\rule{1em}{0ex}}+{\int }_{0.85}^{0.95}Q\left(\begin{array}{c}\hfill -20\phantom{\rule{0.3em}{0ex}}{q}_{2}\left({x}_{2}\right)\hfill \\ \hfill -0.5\phantom{\rule{0.3em}{0ex}}{q}_{2}^{\prime }\left({x}_{2}\right)\hfill \end{array}\right)\cdot \left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}v\left(0.85,{x}_{2}\right)\phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}{x}_{2}.\hfill \end{array}$

The remaining data are

 $\begin{array}{ccc}\hfill {z}_{1}^{″}\left({x}_{1}\right)& =-122\phantom{\rule{1em}{0ex}}880\phantom{\rule{0.3em}{0ex}}{x}_{1}^{4}+122\phantom{\rule{1em}{0ex}}880\phantom{\rule{0.3em}{0ex}}{x}_{1}^{3}-36\phantom{\rule{1em}{0ex}}864\phantom{\rule{0.3em}{0ex}}{x}_{1}^{2}+3\phantom{\rule{1em}{0ex}}072\phantom{\rule{0.3em}{0ex}}{x}_{1}\hfill & \hfill \\ \hfill {z}_{2}^{″}\left({x}_{2}\right)& =-7\phantom{\rule{1em}{0ex}}324.218\phantom{\rule{1em}{0ex}}750\phantom{\rule{0.3em}{0ex}}{x}_{2}^{4}+11\phantom{\rule{1em}{0ex}}718.75\phantom{\rule{0.3em}{0ex}}{x}_{2}^{3}-5\phantom{\rule{1em}{0ex}}625\phantom{\rule{0.3em}{0ex}}{x}_{2}^{2}+750\phantom{\rule{0.3em}{0ex}}{x}_{2}^{1}\hfill \\ \hfill {q}_{1}^{\prime }\left({x}_{1}\right)& =-400\phantom{\rule{0.3em}{0ex}}\left({x}_{1}-0.8\right)\hfill \\ \hfill {q}_{2}^{\prime }\left({x}_{2}\right)& =-400\phantom{\rule{0.3em}{0ex}}\left({x}_{2}-0.9\right).\hfill \end{array}$

### References

C. Meyer and O. Thoma. A priori ﬁnite element error analysis for optimal control of the obstacle problem. SIAM Journal on Numerical Analysis, 51(1):605–628, 2013. doi: 10.1137/110836092.