## ccquas1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via boundary values 1st order

Differential operator:

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

Design constraints:

• box of order 0

State constraints:

• none

Mixed constraints:

• none

Submitted on 2013-02-05 by Roland Herzog. Published on 2013-02-12

## ccquas1 description:

### Introduction

This is a boundary optimal control problem governed by a quasilinear elliptic equation. Problems involving quasilinear equations are particularly challenging with regard to their analysis, numerical analysis, and numerical solution.

This problem and the analytical example were published as the ﬁrst example in Casas and Dhamo  and in [Dhamo2012, Example 4.14].

### Variables & Notation

#### Given Data

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

Note that the formula for the auxiliary term $e\left(x\right)$ above has been corrected (a square was missing). The formula originally given in [Casas and Dhamo2012, p.753] and [Dhamo2012, p.117] does not match the function shown in [Dhamo2012, Figure 4.1]. Preference has been given to the reproduction of the ﬁgure.

### Optimality System

The following optimality system for the state $y\in {H}^{3∕2}\left(\Omega \right)$, the control $u\in {L}^{2}\left(\Gamma \right)$ and the adjoint state $\phi \in {H}^{3∕2}\left(\Omega \right)$, given in the strong form, represents a set of ﬁrst-order necessary optimality conditions.

### Supplementary Material

The following state, control and adjoint state variables are shown in Casas and Dhamo  to satisfy ﬁrst-order necessary conditions. Moreover, second-order suﬃcient conditions also hold due to the structure of the objective. Consequently, $u$ is a local minimum (in the sense of ${L}^{\infty }\left(\Gamma \right)$).

This solution is particular in the sense that the upper and lower bound constraints for the control are both active on nontrivial parts of the boundary, but never strongly active. In other words, strict comlementarity does not hold for this problem. A plot of the optimal control is provided in [Dhamo2012, Figure 4.1].

### References

E. Casas and V. Dhamo. Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations. Computational Optimization and Applications. An International Journal, 52(3): 719–756, 2012. ISSN 0926-6003. doi: 10.1007/s10589-011-9440-0.

V. Dhamo. Optimal Boundary Control of Quasilinear Partial Diﬀerential Equations: Theory and Numerical Analysis. PhD thesis, Technische Universität Berlin, 2012. URL http://opus.kobv.de/tuberlin/volltexte/2012/3511.