# Actuator design for parabolic distributed parameter systems with the moment method

## TL;DR

This paper develops a spectral optimal design method for actuators in parabolic PDEs, optimizing both location and shape using the moment method, with theoretical guarantees and numerical validation.

## Contribution

It introduces a novel spectral optimization framework for actuator shape and placement in parabolic PDEs using the moment method, including existence, uniqueness, and computational procedures.

## Key findings

- Optimal actuator distributions exist and are unique under certain conditions.
- The method effectively maximizes control energy over random initial data.
- Numerical simulations confirm the theoretical results and applicability.

## Abstract

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.