Bounded control of time-delay nonlinear PDEs via an LQR approach: Application to the generalized Kuramoto–Sivashinsky equation

Abstract

This work develops a Linear Quadratic Regulator (LQR)-based optimal feedback controller for nonlinear partial differential equations (PDEs) subject to an input time delay. The controller is designed using the linearized dynamics and is shown to stabilize the full nonlinear system under appropriate assumptions, including the existence of a Fréchet-differentiable semigroup whose derivative coincides with the semigroup generated by the linearization. It is further established that the resulting control law is finite dimensional and provides local stabilization of the infinite dimensional solution, assuming suitable actuator conditions and a finite number of unstable eigenvalues in the linearized model. The effectiveness of the proposed approach is demonstrated on the generalized Kuramoto–Sivashinsky equation (GKS) equation, where the controller successfully stabilizes an unstable flow regime.