Lyapunov-based Boundary Control for a MIMO Counter-Propagating Raman Amplifier

Abstract

This paper considers a boundary feedback control problem for a MIMO counter-propagating Raman amplifier. The system is modeled as a set of coupled semilinear hyperbolic partial differential equations with Lotka-Volterra type nonlinearity. The system is linearized about the steady-state solution, and a boundary controller is designed based on a Lyapunov functional whose time derivative is made strictly negative by an appropriate choice of boundary conditions. As a result, exponential convergence to the steady-state solution is shown in the L2-norm. The results are extended to the nonlinear system under a key assumption.