2021/01/11 by Paul Alphonse, Alphonse, Paul, Jérémy Martin +1
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · doi:10.48550/arxiv.2101.03772
We prove that the thickness property is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space \mathbb Rn. These equations are associated with operators of the form F(\vert Dx\vert), the function F:[0,+∞)→\mathbb R being continuous and bounded from below. We also provide explicit feedbacks and constants associated with these stabilization properties. The notion of thickness is known to be a necessary and sufficient condition for the null-controllability of the fractional heat equations associated with the functions F(t) = t2s in the case s>1/2. Our results apply in particular for this class of equations, but also for the half heat equation associated with the function F(t) = t, which is the most diffusive fractional heat equation for which null-controllability is known to fail from general thick control supports.