Sufficient conditions for elliptic problem of optimal cnotrol in Rn, where n>2

Abstract

This paper is concerned with the local minimization problem for a variety of non Frechet-differentiable Gâteaux functional $J(f)\equiv ın{t}_{Q}v(x,u,f)dx$ in the Sobolev space $(W_0^{1,2}(Q),\Vert \cdot {\Vert }_p)$, where $u$ is the solution of the Dirichlet problem for a linear uniformly elliptic operator with nonhomogenous term $f$ and $\Vert \cdot {\Vert }_p$ 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.