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

Actuator design for parabolic distributed parameter systems with the moment method