Baozhu Guo | Searcharxiv

arXiv2026

Null Controllability for Degenerate Parabolic Equations with Internal Control Applied on a Measurable Subset

This work serves as a continuation of our preceding paper [28]. In that study, we presented a separable variable method to derive the Lebeau-Robbiano spectral inequality for a specific degenerate parabolic equation and subsequently employed it to demonstrate the null controllability of said equation when internal control is applied to an open subset. In the current paper, we reapply the separable variable method to attain the Lebeau-Robbiano spectral inequality for a different degenerate parabolic equation, and we substantiate the null controllability of this equation with internal control acting on a measurable subset. This approach may offer an alternative means of proving controllability results for degenerate parabolic equations.

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arXiv2023

Frequency Energy Multiplier Approach to Uniform Exponential Stability Analysis of Semi-discrete Scheme for a Schrodinger Equation under Boundary Feedback

In this paper, we investigate the uniform exponential stability of a semi-discrete scheme for a Schrödinger equation under boundary feedback stabilizing control in the natural state space $L^2(0,1)$. This study is significant since a time domain energy multiplier that allows proving the exponential stability of this continuous Schrödinger system has not yet found, thus leading to a major mathematical challenge to semi-discretization of the PDE, an open problem for a long time. Although the powerful frequency domain energy multiplier approach has been used in proving exponential stability for PDEs since 1980s, its use to the \emph{uniform} exponential stability of the semi-discrete scheme for PDEs has not been reported yet. The difficulty associated with the uniformity is that due to the parameter of the step size, it involves a family of operators in different state spaces that need to be considered simultaneously. Based on the Huang-Prüss frequency domain criterion for uniform exponential stability of a family of $C_0$-semigroups in Hilbert spaces, we solve this problem for the first time by proving the uniform boundedness for all the resolvents of these operators on the imaginary axis. The proof almost exactly follows the procedure for the exponential stability of the continuous counterpart, highlighting the advantage of this discretization method.

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