OBSERVABILITY AND NULL-CONTROLLABILITY FOR PARABOLIC EQUATIONS IN L-p-SPACES

We study (cost-uniform approximate) null-controllability of parabolic equations in L-p(R-d) and provide explicit bounds on the control cost. In particular, we consider systems of the form <(x)over dot>(t) = -A(p)x(t) + 1(E)u(t), x(0) = x(0) is an element of L-p (R-d), with interior control on a so-called thick set E subset of R-d, where p is an element of [1, infinity), and where A is an elliptic operator of order m is an element of N in L-p(R-d). We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the case p = 1.