Summary: We study internal null controllability for degenerate parabolic equations of Grushin-type \(G_\gamma = \partial_{x x}^2 + | x|^{2 \gamma} \partial_{y y}^2\) \((\gamma > 0)\), in the rectangle \((x, y) \in \Omega = (- 1, 1) \times(0, 1)\). Previous works proved that null controllability holds for weak degeneracies (\(\gamma\) small), and fails for strong degeneracies (\(\gamma\) large). Moreover, in the transition regime and with strip shaped control domains, a positive minimal time is required.
In this paper, we work with controls acting on two strips, symmetric with respect to the degeneracy. We give the explicit value of the minimal time and we characterize some initial data that can be steered to zero in time \(T\) (when the system is not null controllable): their regularity depends on the control domain and the time \(T\).
We also prove that, with a control that acts on one strip, touching the degeneracy line \(\{x = 0 \}\), then Grushin-type equations are null controllable in any time \(T > 0\) and for any degeneracy \(\gamma > 0\).
Our approach is based on a precise study of the observability property for the one-dimensional heat equations satisfied by the Fourier coefficients in variable \(y\). This precise study is done, through a transmutation process, on the resulting one-dimensional wave equations, by lateral propagation of energy method.