MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
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.

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

MSC: 49K20

Pages:  107--119     

Volume  55 ,  Issue  3$-$4 ,  2003