  We consider a shape optimization problem written in the optimal control form:
the governing operator is the $p$-Laplacian in the Euclidean space $\R^d$, the
cost is of an integral type, and the control variable is the domain of the
state equation. Conditions that guarantee the existence of an optimal domain
will be discussed in various situations. It is proved that the optimal domains
have a finite perimeter and, under some suitable assumptions, that they are
open sets. A crucial difference is between the case $p>d$, where the existence
occurs under very mild conditions, and the case $p\le d$, where additional
assumptions have to be made on the data.
