Mengze Gu | Search arXiv

arXiv2026

Optimal damping design and exponential stabilization for weakly degenerate wave equations

We investigate the optimal damping design and exponential stabilization for a class of higher-dimensional weakly degenerate damped wave equations. In weighted Sobolev spaces, we establish well-posedness and uniform exponential energy decay, and verify the existence of long-time optimal damping potentials. To overcome analytical difficulties arising from system non-self-adjointness, we use finite-time approximation via semigroup operator norms. Through linearization and adjoint-state analysis, we derive first-order necessary optimality conditions and show finite-time optimal dampings possess a bang-bang structure.

math.AP↗

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.

math.OC↗