Asymptotic behavior of Carleman weight functions and application to controllability

In the development of controllability and inverse problem results for semi-discrete systems, by using Carleman estimates, it is required to estimate of the discrete operators applied to Carleman weight functions. This work aims to establish the asymptotic behavior of Carleman weight functions under these discrete operators. We provide a characterization of the error term in arbitrary order and dimension, extending previously known results. This generalization is of independent interest due to its applications in deriving Carleman estimates for semi-discrete stochastic operators. The aforementioned estimates hold for Carleman weight functions used for parabolic, hyperbolic, and elliptic operators, which are applied to obtain control and inverse problems results for those operators. We apply these results to obtain $ϕ$-controllability result for a fully discrete parabolic operator, which is based on a Carleman estimate for a fully-discrete parabolic operator.

💡 Research Summary

The paper addresses a fundamental obstacle in the analysis of semi‑discrete and fully discrete partial differential equations (PDEs) used for control and inverse problems: the lack of precise estimates for discrete operators acting on Carleman weight functions. While Carleman estimates are a cornerstone for establishing observability inequalities and, via duality, null‑controllability in the continuous setting, their direct translation to discrete schemes fails in general. The authors first illustrate this failure by constructing a non‑observable eigenfunction for a fully discrete heat equation on a uniform grid, showing that classical null‑controllability cannot be expected for fully discrete systems.

To overcome this limitation, the paper adopts the notion of φ‑null controllability. Instead of demanding exact null‑state at the final time, one seeks controls whose norm remains uniformly bounded with respect to the discretization parameters, while the final state decays at a rate φ(h) that is strictly positive as h→0 (typically of exponential type). This concept has been explored in one‑dimensional settings, but a rigorous multidimensional framework was missing.

The core technical contribution is a systematic asymptotic analysis of the discrete average operator
(A_i u(x)=\frac12\bigl(u(x+\frac{h}{2}e_i)+u(x-\frac{h}{2}e_i)\bigr))
and the discrete difference operator
(D_i u(x)=\frac{1}{h}\bigl(u(x+\frac{h}{2}e_i)-u(x-\frac{h}{2}e_i)\bigr)).
Lemma 2.1 provides product rules for these operators, and Proposition 2.2 extends them to arbitrary powers, yielding a discrete Leibniz formula. Proposition 2.4 and Corollary 2.5 give precise expansions: for any smooth function f,
(A_i^n f = f + R_{A_i^n}(f)),
(D_i^n f = \partial_i^n f + R_{D_i^n}(f)),
where the remainders are of order (h^{2n}) and involve higher‑order derivatives of f. Importantly, these formulas hold in any spatial dimension and for mixed directions, allowing the authors to control cross‑terms that appear when applying several operators in different coordinate directions.

With these discrete expansions in hand, the authors turn to the Carleman weight functions. They consider a weight of the form
(r = e^{s\varphi},\quad \rho = r^{-1}),
where (\varphi = e^{\lambda\psi}) and ψ satisfies the usual geometric conditions (non‑degeneracy, appropriate boundary behavior). Lemma 3.1 and Corollary 3.2 provide continuous‑variable estimates for expressions such as (\partial^\beta(r,\partial^\alpha\rho)), showing they are bounded by (O(\lambda^{s|\alpha|})). The main novelty lies in Theorems 3.5, 3.7, and 3.13, where the authors combine the discrete asymptotics with the continuous Carleman calculus to obtain estimates for discrete operators applied to the weight, e.g.
(A_i^n(r,D_i^m\rho) = r,\partial_i^m\rho + O(h^2\lambda^{s|\alpha|})).
These results generalize the one‑dimensional findings of