## Invited Session Thu.2.MA 415

#### Thursday, 13:15 - 14:45 h, Room: MA 415

**Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods** [...]

### Theory and methods for PDE-constrained optimization problems with inequalities

**Chair: Michael Ulbrich**

**Thursday, 13:15 - 13:40 h, Room: MA 415, Talk 1**

**Francisco José Silva Alvarez**

Characterization of quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

**Coauthors: Terence Bayen, Frédéric Bonnans**

**Abstract:**

In this work, we are concerned with the following optimal control problem:

min_{u} J(u) := ∫_{Ω} *l*(x,y_{u}(x),u(x)) d x,

under bounds constraints on the control *u*, and where *y*_{u} is the unique solution of

\begin{cases}

-Δ y(x) + \varphi(x,y(x),u(x)) = 0, \mathrm{for} x ∈ Ω,\

\hspace{3.75cm} y(x) = 0, \hspace{0.12cm} \mathrm{for} x ∈ \partial Ω.

\end{cases}

We extend to *strong solutions* classical second order analysis results, which are usually established for *weak solutions*. We mean by strong solution a control *\bar{u}* that satisfies:

There exists *ε>0* such that *J(\bar{u**) ≤ J(u)* for all *u* with *|| y*_{u}-y_{\bar{u}}|| ≤ ε.}

The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of the optimization of elliptic equations. Our main result is a characterization of local quadratic growth for the cost function *J* around a strong minimum.

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**Thursday, 13:45 - 14:10 h, Room: MA 415, Talk 2**

**Martin Weiser**

Goal-oriented estimation for nonlinear optimal control problems

**Abstract:**

In optimal control problems with elliptic PDE constraints,

min J(y,u) s.t.* c(y,u) = 0,*

the value of the cost functional is a natural quantity of interest for goal-oriented

error estimation and mesh refinement. The talk will discuss the difference between

the all-at-once error quantity *J(y*^{h},u^{h})-J_{\mathrm{opt}} introduced by Becker/\allowbreak Kapp/\allowbreak Rannacher

and the black-box error quantity *J(y(u*^{h}),u^{h})-J_{\mathrm{opt}}. Both qualitative and

quantitative differences will be addressed for linear-quadratic problems.

In the second part, the black-box approach will be extended to smooth nonlinear problems

and will result in a novel accuracy matching for inexact Newton methods. Quantitative aspects

are illustrated on numerical examples including interior point regularizations of inequality constrained problems.

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**Thursday, 14:15 - 14:40 h, Room: MA 415, Talk 3**

**Florian Kruse**

An infeasible interior point method for optimal control problems with state constraints

**Coauthor: Michael Ulbrich**

**Abstract:**

We present an infeasible interior point method for pointwise state

constrained optimal control problems with elliptic PDEs. A smoothed

constraint violation functional is used to develop a self-concordant

barrier approach in an infinite-dimensional setting.

For the resulting algorithm we provide a detailed convergence analysis

in function space. This includes a rate of convergence and a rigorous

measure for the proximity of the actual iterate to both the path of

minimizers and the solution of the problem.

Moreover, we report on numerical experiments to illustrate the

efficiency and the mesh independence of this algorithm.