Local exponential stabilisation of the delayed Fisher's equation

Exponential stabilisation is undoubtedly an essential property to study for any control system. This property is often difficult to establish for nonlinear distributed systems and even more difficult if the state of such a system is subject to a time-delay. In this context, this paper is devoted to the study of the well-posedness and exponential stabilisation problems for the one-dimensional delayed Fisher's partial differential equation (PDE) posed in a bounded interval, with homogeneous Dirichlet condition at the left boundary and controlled at the right boundary by Dirichlet condition. Based on the infinite dimensional backstepping method for the linear and undelayed corresponding case, a feedback law is built. Under this feedback, we prove that the closed-loop system is well-posed in $ L^2(0,1) $ L2(0,1) and locally exponentially stable in the topology of the $ L^2(0,1) $ L2(0,1)-norm by the use of semigroup theory relating to differential difference equations and Lyapunov–Razumikhin analysis, respectively. A numerical example that reflects the effectiveness of the proposed approach is given.