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

**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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*Talk 1 of the invited session Thu.2.MA 415

**"Theory and methods for PDE-constrained optimization problems with inequalities"** [...]

Cluster 19

**"PDE-constrained optimization & multi-level/multi-grid methods"** [...]