Observability and null-controllability for parabolic equations in Lp-spaces

We study (cost-uniform approximate) null-controllability of parabolic equations in Lp(Rd ) and provide explicit bounds on the control cost. In particular, we consider systems of the form ẋ(t) = −Apx(t) + 1E u(t), x(0) = x0 ∈ Lp(Rd ), with interior control on a so-called thick set E ⊂ Rd, where p ∈ [1, ∞), and where A is an elliptic operator of order m ∈ N in Lp(Rd ). 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.

Subjects

Banach space

C -semigroups 0

elliptic operators

L -spaces p

non-reflexive

Null-controllability

observability estimate

DDC Class

510: Mathematics

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Observability and null-controllability for parabolic equations in Lp-spaces