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Actuator design for parabolic distributed parameter systems with the moment method. (English) Zbl 1361.49020

Summary: 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 \(\mathbb{R}^n\). 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.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
93B07 Observability
35L05 Wave equation
49K20 Optimality conditions for problems involving partial differential equations
42B37 Harmonic analysis and PDEs
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