Approximate controllability for linear degenerate parabolic problems with bilinear control

In this work we study the global approximate multiplicative controllability for the linear degenerate parabolic Cauchy-Neumann problem $$ \left\{\begin{array}{l} \displaystyle{v_t-(a(x) v_x)_x =\alpha (t,x)v\,\,\qquad \mbox{in} \qquad Q_T \,=\,(0,T)\times(-1,1) }\\ [2.5ex] \displaystyle{a(x)v_x(t,x)|_{x=\pm 1} = 0\,\,\qquad\qquad\qquad\,\, t\in (0,T) }\\ [2.5ex] \displaystyle{v(0,x)=v_0 (x) \,\qquad\qquad\qquad\qquad\quad\,\, x\in (-1,1)}~, \end{array}\right. $$ with the bilinear control $\alpha(t,x)\in L^\infty (Q_T).$ The problem is strongly degenerate in the sense that $a\in C^1([-1,1]),$ positive on $(-1,1),$ is allowed to vanish at $\pm 1$ provided that a certain integrability condition is fulfilled. We will show that the above system can be steered in $L^2(\Omega)$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear static controls. Moreover, we extend the above result relaxing the sign constraint on $v_0$.