Local null controllability of a parabolic-elliptic chemotaxis system by a Neumann boundary control

Abstract

We prove local null controllability for a parabolic equation coupled to an elliptic equation on a bounded interval. The drift depends on the elliptic variable, and a single control acts through the Neumann condition at the left endpoint. For every positive control time and every locally Lipschitz drift function, sufficiently small initial data in $L^\infty$ can be driven to zero by an $L^\infty$ boundary control. To construct the control, we extend the linear equation to a larger interval using an even extension of the initial datum and an odd extension of the drift. A distributed $L^2$ control in the added interval then gives a boundary control on the original interval. A heat kernel estimate bounds the derivative trace needed for this construction. We obtain the nonlinear result by a set-valued fixed point argument.