Control Design of Nonlinear ODE/Transport PDE Cascaded Systems with Varying Propagation Speed

We consider control design for a nonlinear ordinary differential equation (ODE) and transport partial differential equation (PDE) cascaded system whose propagation speed is spatially-varying. The ODE state is driven by the uncontrolled boundary of the transport PDE. Our nonlinear delay-compensated control law enables global asymptotic stability of the closed-loop system. The controller design is deduced using a backstepping transformation and constructing a Lyapunov functional. We offer global stability guaranteed by the assumption of the propagation speed is continuously differentiable and positive. The validity of the proposed controller is illustrated by a simulation result.