Sufficient conditions for elliptic problem of optimal control in $R^n$
in Orlicz Sobolev spaces

S. Lahrech and A. Addou

Abstract

This paper is concerned with the local minimization problem for a
variety of non Frechet-differentiable G\^ateaux functional
$J(f)\equiv\int_{\Omega}v(x,u,f)\,dx$ in the Orlicz-Sobolev space
$(W^1_0L_M^*(\Omega),\|.\|_{M})$, where $u$ is the solution of
the Dirichlet problem for a linear uniformly elliptic
operator with nonhomogenous term $f$ and $\|.\|_{M}$ is
the Orlicz norm associated with an N-function~$M$.
We use a recent extension of Frechet-differentiability
(approach of Taylor mappings see [2]), and we give
various assumptions on $v$ to guarantee a critical point is
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.