## gcdist1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Poisson:
• linear elliptic operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• No time dependence.

Design constraints:

• none

State constraints:

• nonlinear convex, local of order 1

Mixed constraints:

• none

Submitted on 2013-01-14 by Winnifried Wollner. Published on 2013-01-14

## gcdist1 description:

### Introduction

This is a variation of the mother problem with additional pointwise constraints on the gradient of the state with known analytic solution. The presented problem is given on a domain $\Omega \subset {ℝ}^{2}$. This problem and analytical solution where proposed in [Deckelnick et al.2008, Section 5], and have been veriﬁed in Wollner [2010]. The solution of the problem is special due to the fact that no additional bounds on the control are needed.

### Variables & Notation

#### Given Data

The given data is chosen in a way which admits an analytic solution, that is given by rotation of a one dimensional problem.

### Optimality System

The following optimality system for the state $y\in {H}_{0}^{1}\left(\Omega \right)\cap {W}^{2,p}\left(\Omega \right)$ with $p>2$, the control $u\in {L}^{2}\left(\Omega \right)$, the adjoint state $p\in {L}^{{p}^{\prime }}\left(\Omega \right)$ where $\frac{1}{p}+\frac{1}{{p}^{\prime }}=1$, and a Lagrange multiplier $\mu \in M{\left(\Omega \right)}^{2}={C}^{\ast }{\left(\overline{\Omega }\right)}^{2}$ for the constraint on the gradient of $y$ characterizes the unique minimizer, see Casas and Fernández [1993]:

Here the adjoint equation has to be understood in the very weak sense, i.e., $p$ solves

$-{\int }_{\Omega }p△\varphi \phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}x={\int }_{\Omega }\left(y-{y}_{\Omega }\right)\varphi \phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}x+{\int }_{\Omega }\nabla \varphi \phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}\mu \phantom{\rule{1em}{0ex}}\forall \varphi \in {H}_{0}^{1}\left(\Omega \right)\cap {C}^{1}\left(\overline{\Omega }\right).$

### Supplementary Material

The optimal state, adjoint state, control and Lagrange multiplier are known analytically:

 $\begin{array}{ccc}\hfill y& ={y}_{\Omega },\hfill & \hfill \\ \hfill p& =-u,\hfill \\ \hfill u& =\left\{\begin{array}{cc}-1,\phantom{\rule{1em}{0ex}}\hfill & 0\le |x|\le 1,\hfill \\ 0,\phantom{\rule{1em}{0ex}}\hfill & 1<|x|\le 2,\hfill \end{array}\right\\hfill \\ \hfill \mu & =\frac{\nabla y}{|\nabla y|}{\mu }_{0},\hfill \\ \hfill {⟨\varphi ,{\mu }_{0}⟩}_{C,{C}^{\ast }}& ={\int }_{|x|=1}\varphi \phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}s.\hfill \end{array}$

### References

E. Casas and L. A. Fernández. Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Applied Mathematics and Optimization, 27:35–56, 1993. doi: 10.1007/BF01182597.

K. Deckelnick, A. Günther, and M. Hinze. Finite element approximation of elliptic control problems with constraints on the gradient. Numerische Mathematik, 111: 335–350, 2008. doi: 10.1007/s00211-008-0185-3.

W. Wollner. A posteriori error estimates for a ﬁnite element discretization of interior point methods for an elliptic optimization problem with state constraints. Computational Optimization and Applications, 47(1):133–159, 2010. doi: 10.1007/s10589-008-9209-2.