Memory-Type Null Controllability for Non-Autonomous Degenerate Parabolic Equations with Boundary Degeneracy

TL;DR

This paper establishes memory-type null controllability for non-autonomous degenerate parabolic equations with boundary degeneracy, using new Carleman estimates and observability inequalities.

Contribution

It introduces novel Carleman estimates for non-autonomous degenerate operators and addresses memory effects in controllability analysis.

Findings

Proves memory-type null controllability under structural conditions.

Develops Carleman estimates for degenerate non-autonomous operators.

Handles memory as a lower-order perturbation in the framework.

Abstract

This paper studies the memory-type null controllability of a class of one-dimensional non-autonomous degenerate parabolic equations with Volterra-type memory terms. The diffusion operator is considered in both divergence and non-divergence forms and may exhibit weak or strong degeneracy at the boundary, while the diffusion coefficient depends explicitly on time. Due to the presence of memory effects, classical null controllability is insufficient, and a stronger notion requiring the vanishing of both the state and the accumulated memory is introduced. To address this problem, we establish new Carleman estimates adapted to non-autonomous degenerate operators in weighted spaces. The memory term is handled as a lower-order perturbation within the Carleman framework. These estimates yield suitable observability inequalities, which allow us to prove memory-type null controllability under appropriate structural conditions. Extensions to cases with double boundary degeneracy and moving control regions are also discussed.