  We provide new uncertainty principles for functions in a general class of
Gelfand-Shilov spaces. These results apply, in particular, with the classical
Gelfand-Shilov spaces as well as for spaces of functions with weighted Hermite
expansions. Thanks to these uncertainty principles, we derive
null-controllability results for evolution equations with adjoint systems
enjoying smoothing effects in specific Gelfand-Shilov spaces. More precisely,
we consider control subsets which are thick with respect to a quasi linearly
growing density and establish sufficient conditions on the growth of the
density to ensure null-controllability of these evolution equations.
