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 = 1E is the restriction onto a thick set E ⊂ Rd. In this case, we show that the above system satisfies a final state observability estimate if and only if E ⊂ 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.