On the Controllability of a Fractional Order Parabolic Equation

The null-controllability property of a 1 − d parabolic equation involving a fractional power of the Laplace operator, (−∆), is studied. The control is a scalar time-dependent function g = g(t) acting on the system through a given space-profile f = f(x) on the interior of the domain. Thus, the control g determines the intensity of the space control f applied to the system, the latter being given a priori. We show that, if α ≤ 1/2 and the shape function f is, say, in L, no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case α > 1/2 and, in particular, for the heat equation that corresponds to α = 1. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation. On the contrary, if more regularity of the shape function f is assumed, then we show that there are initial data in any Sobolev space H that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases. These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when α = 1/2 for the fractional powers of the Dirichlet Laplacian in 1 − d. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [17] and [18]. We also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models.