Null controllability for degenerate parabolic equations with a nonlocal space term

TL;DR

This paper investigates the null controllability of degenerate parabolic equations with nonlocal space terms, establishing well-posedness, Carleman estimates, and applying fixed point theorems to prove controllability.

Contribution

It introduces a novel approach combining Carleman estimates and fixed point theory to achieve null controllability for nonlocal degenerate heat equations.

Findings

Established well-posedness of the nonlocal degenerate heat equations

Derived Carleman estimates for the adjoint problems

Proved null controllability using fixed point arguments

Abstract

We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.