  In this paper, we model and solve the problem of designing in an optimal way
actuators for parabolic partial differential equations settled on a bounded
open connected subset $\Omega$ of IR n. We optimize not only the location but
also the shape of actuators, by finding what is the optimal distribution of
actuators in $\Omega$, over all possible such distributions of a given measure.
Using the moment method, we formulate a spectral optimal design problem, which
consists of maximizing a criterion corresponding to an average over random
initial data of the largest L 2-energy of controllers. Since we choose the
moment method to control the PDE, our study mainly covers one-dimensional
parabolic operators, but we also provide several examples in higher dimensions.
We consider two types of controllers: either internal controls, modeled by
characteristic functions, or lumped controls, that are tensorized functions in
time and space. Under appropriate spectral assumptions, we prove existence and
uniqueness of an optimal actuator distribution, and we provide a simple
computation procedure. Numerical simulations illustrate our results.
