Observability and null-controllability for parabolic equations in $L_p$-spaces

Abstract

We study null-controllability of parabolic equations in Banach spaces. We show that a generalized uncertainty principle and a dissipation estimate imply (approximate) null-controllability. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, we do not assume reflexivity of the underlying Banach space, thus allowing to apply our result, e.g., to L 1 -spaces. As an application we consider parabolic equations of the form ẋ), with interior control on a so-called thick set E ⊂ R d , where p ∈ [1, ∞), and where A is an elliptic operator of order m ∈ N in L p (R d ). In particular, our result applies to the case p = 1.

Final State Observability in Banach spaces with applications to Subordination and Semigroups induced by L{é}vy processes

“…Moreover, let C ∈ L(L p (R n ), L p (E)) be the restriction operator and for λ > 0 let P λ ∈ L(L p (R n )) be induced by a smooth cut-off function 1 B(0,λ/2) ≤ χ λ ≤ 1 B(0,λ) in Fourier space; cf. [BGST21]. Then the uncertainty principle is satisfied by a Logvinenko-Sereda type theorem (see Proposition 4.11 below or [EV18, WWZZ19, BGST21]).…”

“…Then the uncertainty principle is satisfied by a Logvinenko-Sereda type theorem (see Proposition 4.11 below or [EV18, WWZZ19, BGST21]). Moreover, exploiting the heat kernel, we deduce a dissipation estimate with g and h(t) = t for all t > 0 (see [BGST21]). Hence, Theorem 3.8 is applicable and we obtain final state observability for (S ϕ (t)) t≥0 , which is generated by the fractional Laplacian −ϕ(−A) = −(−A) s = −(−∆) s .…”

Sufficient criteria for stabilization properties in Banach spaces

“…Remark 3.3. Let us relate Proposition 3.1 to the results obtained in [HWW21] and [GST20,BGST21]. By choosing the functions C 1 , C 2 : (0, T ] → [0, ∞) appropriately we can mimic the assumptions of [HWW21, Lemma 2.2] and [GST20, Theorem 2.1], respectively.…”

“…One possible approach to prove null-controllability is a method known as Lebeau-Robbiano strategy, originating in the seminal work by Lebeau and Robbiano [LR95], see also [LZ98,JL99]. Subsequently, this strategy was generalised in various steps to C 0 -semigroups on Hilbert spaces, see, e.g., [Mil10,TT11,WZ17,BPS18,NTTV20], and more recently to C 0semigroups on Banach spaces, see [GST20,BGST21]. The essence of this approach is to show an uncertainty principle and a dissipation estimate for the dual system which are valid for an infinite sequence of so-called spectral parameters, and prove that the growth rate in the uncertainty principle is strictly smaller than the decay rate of the dissipation estimate.…”

Observability for Non-autonomous Systems

“…Of particular interest in understanding the case of unbounded domains is the specification of necessary and sufficient geometric conditions on ω for observability, which were established in [EV18,WWZZ19] in the case of Hilbert spaces. In the Banach space case, a characterization of observability in terms of geometric conditions was given in [GST20,BGST].…”