Estimates for the controls of the wave equation with a potential

Micu, Sorin; Temereancă, Laurenţiu Emanuel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 289-309.

This article studies the $L^2$-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential $a=a\left(x\right)$. It is known these controls depend on $a$ and their norms may increase exponentially with $∥a∥{}_{L^{\infty }}$. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal $L^2$−norm controls are uniformly bounded with respect to the potential $a$, if the initial data have only sufficiently high eigenmodes.

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