Null controllability of degenerate parabolic equations with a time-varying delay

Abstract

This paper is devoted to the null controllability of a one-dimensional linear parabolic equation with a time-varying state delay and a diffusion coefficient that degenerates at one endpoint. Both the weakly and strongly degenerate cases are considered. When the delay map $g(t)=t-\tau(t)$ is strictly increasing, the adjoint equation contains an advanced term weighted by the Jacobian associated with $g^{-1}$. For a distributed control acting on an interior subinterval, we combine a degenerate Carleman estimate on the final delay-free interval with a monotonicity functional whose upper integration limit moves in time. The resulting observability inequality gives null controllability under a terminal decay condition on the delayed coefficient outside the control region. We then obtain Dirichlet boundary controllability at the nondegenerate endpoint by extending the equation in space. The constant-delay equation is recovered as a special case.