Interplay between Discretization and Controllability. Example of a Wave Equation with Boundary Control
Résumé
In this paper, we study the interplay between controllability (in particular, spectral controllability) and discretization for the wave equation with boundary control. First, we recall how it can be transformed into an equivalent linear delay system and show how varying polynomial operators allows recovering diverse boundary conditions. Then, by analyzing the delay system associated with the wave equation, we examine controllability properties, focusing on whether controllability is preserved under discretization. We show that, depending on the polynomials defining the boundary conditions, the same discretization scheme can lead to different outcomes: a spectrally controllable continuous-time partial differential equation (viewed as an appropriate delay system) may result in either a controllable discretized system or a non-controllable one, regardless of the discretization period.
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