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


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

minu J(u) := ∫Ω l(x,yu(x),u(x)) d x,

under bounds constraints on the control u, and where yu is the unique solution of

 -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.

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 || yu-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.


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" [...]


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