arXiv subjects

Jingrui Niu

Publications and source records attributed to Jingrui Niu.

8 recordsLinked to original sources

arXiv2026

Stability of a Korteweg--de Vries equation close to critical lengths

In this paper, we investigate the quantitative exponential stability of the Korteweg-de Vries equation on a finite interval with its length close to the critical set. Sharp decay estimates are obtained via a constructive PDE control framework. We first introduce a novel transition-stabilization approach, combining the Lebeau--Robbiano strategy with the moment method, to establish constructive null controllability for the KdV equation. This approach is then coupled with precise spectral analysis and invariant manifold theory to characterize the asymptotic behavior of the decay rate as the length of the interval approaches the set of critical lengths. Building on our classification of the critical lengths, we show that the KdV equation exhibits distinct asymptotic behaviors in neighborhoods of different types of critical lengths.

math.AP↗

arXiv2026

Symmetry conditions for spacetime observability of wave equations on the torus

We study observability for the one-dimensional wave equation on the torus from spacetime measurable observation sets. While the Geometric Control Condition (GCC) provides a sufficient criterion in many classical settings, it is no longer sufficient in this framework. We construct explicit counterexamples showing the failure of observability despite the validity of GCC. This leads to the introduction of an additional symmetry condition on the observation set, referred to as the Observable Symmetry Condition (OSC). We prove that observability holds if and only if both GCC and OSC are satisfied. We also show that unique continuation holds if and only if both OSC and a weak form of GCC are satisfied.

math.AP↗

arXiv2025

Small-time local controllability of a KdV system for all critical lengths

In this paper, we consider the small-time local controllability problem for the KdV system on an interval with a Neumann boundary control. In 1997, Rosier discovered that the linearized system is uncontrollable if and only if the length is critical, namely $L=2π\sqrt{(k^2+ kl+ l^2)/3}$ for some integers $k$ and $l$. Coron and Crépeau (2003) proved that the nonlinear system is small-time locally controllable even if the linearized system is not, provided that $k= l$ is the only solution pair. Later, Cerpa and Crepeau showed that the system is large-time locally controllable for all critical lengths. In 2020, Coron, Koenig, and Nguyen found that the system is not small-time locally controllable if $2k+l\not \in 3\mathbb{N}^*$. We demonstrate that if the critical length satisfies $2k+l \in 3\mathbb{N}^*$ with $k\neq l$, then the system is not small-time locally controllable. This paper, together with the above results, gives a complete answer to the longstanding open problem on the small-time local controllability of KdV on all critical lengths since the pioneer work by Rosier

math.AP↗

arXiv2025

The periodic KdV with control on space-time measurable sets

In this paper, we establish the local exact controllability of the KdV equation on torus around equilibrium states, where both the spatial control region and the temporal control region are sets of positive measure. The proof is based on a novel strategy for proving observability inequalities on space-time measurable sets. This approach is applicable to a broad class of dispersive equations on torus.

math.AP↗

arXiv2025

Geometric condition for the observability of electromagnetic Schr\"odinger operators on $\mathbb{T}^2$

In this article we revisit the observability of the Schr\"odinger equation on the two-dimensional torus. In contrast to the Schr\"odinger operator with a purely electric potential, for which any non-empty open set guarantees observability, the presence of a magnetic potential introduces an additional obstruction. We establish a sufficient and almost necessary geometric condition for the observability of electromagnetic Schr\"odinger operators. This condition incorporates the magnetic potential, which can also be characterized by a geometric control condition for the corresponding magnetic field.

math.AP↗

arXiv2025

Observability and controllability for Schrödinger equations in the semi-periodic setting

Strichartz estimates, well-posedness theory and long time behavior for (nonlinear) Schrödinger equations on waveguide manifolds $\mathbb{R}^m \times \mathbb{T}^n$ are intensively studied in recent decades while the corresponding control theory and observability estimates remain incomplete. The purpose of this short paper is to investigate the observability and controllability for Schrödinger equations in the waveguide (semi-periodic) setting. Our main result establishes local exact controllability for the cubic nonlinear Schrödinger equations (NLS) on $\mathbb{R}^2 \times \mathbb{T}$, under certain geometric conditions on the control region. To address the nonlinear control problem, we begin by analyzing the observability properties of the linear Schrödinger operator on a general waveguide manifold $\mathbb{R}^m \times \mathbb{T}^n$. Utilizing $H^s$ estimates of the Hilbert Uniqueness Method (HUM) operator and Bourgain spaces, we then prove local exact controllability through a fixed-point method.

math.AP↗

arXiv2023

Controllability of quasi-linear Hamiltonian Schrödinger equations on tori

We prove exact controllability for quasi-linear Hamiltonian Schrödinger equations on tori of dimension greater or equal then two. The result holds true for sufficiently small initial conditions satisfying natural minimal regularity assumptions, provided that the region of control satisfies the geometric control condition.

math.AP↗

arXiv2020

Simulataneous Control of Wave Systems

In this paper, we study the simultaneous controllability of wave systems in an open domain of R d , d $\in$ N *. We obtain a partial controllability result on a co-finite dimensional space for wave equations coupled by a single control function. We use microlocal defect measures and the unique continuation property of eigenfunctions to prove that an appropriate observability inequality holds for wave equations with space varying and different speeds coupled by a single control function. For the unique continuation property of eigenfunctions, we construct a counterexample to show that in some metrics, the unique continuation property does not hold. Moreover, we study different conditions to ensure the unique continuation property. We also extend our result to the case of constant coefficients and possibly multiple control functions. In this context, we prove the controllability property is equivalent to an appropriate Kalman rank condition.

math.AP↗