Sufficient conditions for elliptic problem of optimal cnotrol in
$R^n$, where $n>2$

S. Lahrech, A. Addou

Abstract

This paper is concerned with the local minimization problem for a
variety of non Frechet-differentiable G\^ateaux functional
$J(f)\equiv \int_{Q}v(x,u,f)\,dx$ in the Sobolev space
$(W^{1,2}_0(Q),\|\cdot\|_p)$, where $u$ is the solution of
the Dirichlet problem for a linear uniformly elliptic
operator with nonhomogenous term $f$ and $\|\cdot\|_{p}$ is
the norm generated by the metric space $L^p(Q)$,
$(p>1)$. We use a recent extension of
Frechet-differentiability (approach of Taylor mappings, see [5]),
and we give various assumptions on $v$ to guarantee a critical point to be
a strict local minimum. Finally, we give an example of a control problem where
classical Frechet differentiability cannot be used and their approach of
Taylor mappings works.

Keywords: Elliptic problem, optimal control, local minimization, Dirichlet
problem, Frechet-differentiability.