Null controllability of degenerate parabolic equations with a time-varying delay
arXiv:1902.01907
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 is strictly increasing, the adjoint equation contains an advanced term weighted by the Jacobian associated with . 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.
Substantially revised and extended version. The constant-delay framework has been generalized to a time-varying delay. The adjoint formulation, observability argument, and boundary controllability result have been revised accordingly. Title changed to reflect the broader scope. 14 pages