Null controllability for parabolic operators with interior degeneracy and one-sided control

For alpha is an element of (0, 2) we study the null controllability of the parabolic operator Pu = u(t) - (vertical bar x vertical bar(alpha)ux)(x) (-1 < x < 1), which degenerates at the interior point x = 0 for locally distributed controls acting only on one side of the origin (that is, on some interval (a, b) with 0 < a < b < 1). Our main results guarantee that P is null controllable if and only if it is weakly degenerate, that is, alpha is an element of (0, 1). (So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case alpha is an element of [1, 2).) Our approach is based on spectral analysis and the moment method. We also provide numerical evidence to illustrate our theoretical results.