On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

On the absence of trapped modes in locally perturbed open waveguides

“…By the inverse generalized Fourier transform, u vanishes outside a section containing the support of f. From the unique continuation principle [15] applied to equation (20), we conclude that u vanishes outside the support of f. This completes the proof of Theorem 5, and thus also of Theorem 1.…”

“…Theorem 1 is actually a corollary of the latter: it suffices to apply the unique continuation principle [15] to equation (1), which tells us that u also vanishes inside the support of f.…”

Null controllability of degenerate parabolic equations of Grushin and Kolmogorov type

“…Our construction of the control steering the initial state to zero is the one of [8], which is in turn inspired by [33] (see also [34]). …”

“…For the proof of Proposition 8 we shall need the following inequality obtained in [33] (see also [34]). …”

Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences

“…Carleman inequalities have been introduced by [7] for proving uniqueness results for some PDE's and have been widely extended by Hörmander (see [16,17]). See also [23] where different Carleman inequalities are presented and compared and where some applications to the controllability of the heat equation is also done.…”