Boundary Control of Nonlinear ODE/Wave PDE Systems With a Spatially Varying Propagation Speed

Abstract

We consider the boundary control of a nonlinear ordinary differential equation (ODE) actuated through a wave equation whose propagation speed is spatially varying. The ODE state is driven by the uncontrolled boundary of the wave equation. We design a nonlinear backstepping compensator to enable global asymptotic stability of the closed-loop system. We deduce the controller design and the stability proof by introducing a two-step backstepping transformation. The first transformation recasts the original system into a coupled 2×2 first-order hyperbolic system with spatially varying coefficients cascading into a nonlinear ODE. The second transformation is used in the design of a compensator for the resulting cascaded system. Our design offers a global stability result that is guaranteed assuming that the spatially varying propagation speed is continuously differentiable and positive. Moreover, for nonlinear systems, our result is the first contribution enabling actual compensation of actuator delays governed by a coupled first-order hyperbolic partial differential equation (PDE) induced by a wave PDE dynamics with a spatially varying propagation speed. The validity of the proposed controller is illustrated by the benchmark system controlled via a cable.