Local exponential stabilization of Fisher’s equation using the backstepping technique

This paper deals with the boundary stabilization problem for the 1D Fisher’s partial differential equation (PDE) defined on a bounded interval, which is a nonlinear unstable distributed parameter system. The stabilizing controller derived using the backstepping technique for linear parabolic PDEs (associated with Fisher’s PDE) can be extended to the nonlinear case. In the work, we can prove that the boundary controller based on the linear backstepping synthesis approach can make the closed-loop nonlinear Fisher’s system exponentially stable (with any desired decay rate) locally in both L2(0,1)L2(0,1) and H1(0,1)H1(0,1), respectively. The effectiveness of the control synthesis approach is illustrated by a numerical simulation.