Logarithmic Convexity and impulse Approximate Controllability for Degenerate Parabolic Equations with Robin Boundary Conditions

TL;DR

This paper proves approximate controllability for certain degenerate parabolic equations with Robin boundary conditions using a logarithmic convexity approach and impulsive control, contributing new insights into control theory for degenerate PDEs.

Contribution

It introduces a novel application of logarithmic convexity and Carleman commutator techniques to establish controllability for degenerate parabolic equations with Robin boundaries.

Findings

Established an observability inequality for the adjoint system.

Proved approximate controllability at the final time.

Applied impulsive control in a localized region.

Abstract

In this work, we investigate the approximate controllability of a class of one-dimensional degenerate parabolic equations with Robin boundary conditions. The degeneracy occurs at one endpoint of the spatial domain, and we apply an impulsive control in a small region at a fixed moment. Our main result establishes an observability inequality for the adjoint system, from which we deduce approximate controllability at final time . The proof relies on a logarithmic convexity argument, developed through a Carleman commutator approach.