LACK OF NULL-CONTROLLABILITY FOR THE FRACTIONAL HEAT EQUATION AND RELATED EQUATIONS

We consider the equation (partial derivative(t) rho(root-Delta))f (t, x) = 1(omega)u(t,x), x is an element of R or T. We prove it is not null-controllable if rho is analytic on a conic neighborhood of R+ and rho(xi) = o(vertical bar xi vertical bar). The proof relics essentially on geometric optics, i.e., estimates for the evolution of semiclassical coherent states. The method also applies to other equations. The most interesting example might be the Kolmogorov-type equation (partial derivative(t) - partial derivative(2)(v) + v(2)partial derivative(x))f (t, x, v) = 1(omega)u(t, x, v) for (x, v) is an element of Omega(x) x Omega(v) with Omega(x) = R or T and Omega(v) = R or (-1, 1). We prove it is not null-controllable in any time if omega is a vertical band omega(x) x Omega(v). The idea is to note that, for some families of solutions, the Kolmogorov equation behaves like the rotated fractional heat equation (partial derivative(t) +root i(-Delta)(1/4))g(t, x) = 1(omega)u(t, x), x is an element of T.