Sufficient criteria and sharp geometric conditions for observability in Banach spaces

Abstract

Let X,Y be Banach spaces, (St)t ≥ 0 a C0-semigroup on X, -A the corresponding infinitesimal generator on X, C a bounded linear operator from X to Y, and T > 0. We consider the system \[ x(t) = -Ax(t), y(t) = Cx(t) t∈ (0,T], x(0) = x0 ∈ X. \] We provide sufficient conditions such that this system satisfies a final state observability estimate in Lr ((0,T) ; Y), r ∈ [1,∞]. These sufficient conditions are given by an uncertainty relation and a dissipation estimate. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider the example where A is an elliptic operator in Lp(Rd) for 1<p<∞, and where C = 1ω is the restriction onto a thick set ω ⊂ Rd. In this case, we show that the above system satisfies a final state observability estimate if and only if ω ⊂ Rd is a thick set. Finally, we make use of the well-known relation between observability and null-controllability of the predual system, and investigate bounds on the corresponding control costs.