The cost of controlling degenerate parabolic equations by boundary controls

We consider the one-dimensional degenerate parabolic equation $$ u_t - (x^\alpha u_x)_x =0 \qquad x\in(0,1),\ t \in (0,T) ,$$ controlled by a boundary force acting at the degeneracy point $x=0$. First we study the reachable targets at some given time $T$ using $H^1$ controls, extending the moment method developed by Fattorini and Russell to this class of degenerate equations. Then we investigate the controllability cost to drive an initial condition to rest, deriving optimal bounds with respect to $\alpha$ and deducing that the cost blows up as $\alpha \to 1^-$.