## scdist1 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:

• box of order 0

Mixed constraints:

• none

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

## scdist1 description:

### Introduction

This problem is a standard linear-quadratic optimal control problem with pointwise state constraints, but no constraints on the control variable. It appears in Meyer et al. [2005] as a test example for the so-called Lavrentiev regularization approach. By the latter, optimal control problems with state constraints of the form $y\left(x\right)\ge {y}_{c}\left(x\right)$ are approximated by problems involving mixed control-state constraints $y\left(x\right)+𝜀\phantom{\rule{0.3em}{0ex}}u\left(x\right)\ge {y}_{c}\left(x\right)$. While problems with pointwise state constraints usually involve a measure-valued Lagrange multiplier, see Casas [1986], the corresponding quantity in the regularized problem is a function.

The following state-constrained problem and its solution appear in [Meyer et al.2005, Section 7, Example 2] and it features a line measure as the state constraint multiplier.

### Variables & Notation

#### Given Data

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

Note that there is a typo in the speciﬁcation of ${y}_{d}$ in [Meyer et al.2005, Section 7, Example 2].

### Optimality System

The following optimality system for the state $y\in {H}^{1}\left(\Omega \right)$, the control $u\in {L}^{2}\left(\Omega \right)$, the adjoint state $p\in {H}^{1}\left(\Omega \right)$ and the state constraint multiplier $\mu \in M\left(\overline{\Omega }\right)$, given in the strong form, characterizes the unique minimizer.

The space $M\left(\overline{\Omega }\right)$ is the dual of $C\left(\overline{\Omega }\right)$ and it consists of all signed, real, regular Borel measures on $\overline{\Omega }$. Here, ${\mu }_{\Omega }$ and ${\mu }_{\partial \Omega }$ denote the restrictions of the measure $\mu$ to $\Omega$ and its boundary, respectively.

### Supplementary Material

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

Note that $\mu$ consists of a regular part (the characteristic function of ${\Omega }_{2}$) plus a singular part (the line measure concentrated on $\stackrel{̂}{\Gamma }$). In particular, the boundary part ${\mu }_{\partial \Omega }$ vanishes.

### Revision History

• 2014–11–14: added missing term ${u}_{d}$ to the objective (thanks to Stefan Takacs)
• 2012–11–14: problem added to the collection

### References

E. Casas. Control of an elliptic problem with pointwise state constraints. SIAM Journal on Control and Optimization, 24(6):1309–1318, 1986. doi: 10.1137/0324078.

C. Meyer, A. Rösch, and F. Tröltzsch. Optimal control of PDEs with regularized pointwise state constraints. Computational Optimization and Applications, 33(2–3): 209–228, 2005. doi: 10.1007/s10589-005-3056-1.