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  vt − (a(x)vx)x = α(t, x)v in QT = (0, T )× (−1, 1) a(x)vx(t, x)|x=±1 = 0 t ∈ (0, T ) v(0, x) = v0(x) x ∈ (−1, 1) , with the bilinear control α(t, x) ∈ L(QT ). The problem is strongly degenerate in the sense that a ∈ C([−1, 1]), positive on (−1, 1), is allowed to vanish at ±1 provided that a certain integrability condition is fulfilled. We will show that the above system can be steered in L(Ω) 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 v0.