Actuator design for parabolic distributed parameter systems with the moment method

First Published in SIAM Journal on Control and Optimization in Volume 55, Issue 2, 2017, Pages 1128-1152, published by the Society for Industrial and Applied Mathematics (SIAM)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 of Rn. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in , 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 L2-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 resultsThe first author was partially supported by ANR project OPTIFORM. This work was partially supported by Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, GA 694126, ICON of the French ANR-2016-ACHN-0014-01, FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of EOARD-AFOSR, and grant MTM2014-52347 of MINECO (Spain

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An approach to optimal actuator design based on shape and topology optimisation techniques is presented. For linear diffusion equations, two scenarios are considered. For the first one, best actuators are determined depending on a given initial condition. In the second scenario, optimal actuators are determined based on all initial conditions not exceeding a chosen norm. Shape and topological sensitivities of these cost functionals are determined. A numerical algorithm for optimal actuator design based on the sensitivities and a level-set method is presented. Numerical results support the proposed methodology.Comment: 41 pages, several figure

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