  In this paper, we are concerned with the state feedback stabilization of
ODE-PDE cascade systems governed by a linear ordinary differential equation and
the 1-d reaction-diffusion equation posed on a bounded interval. In contrast to
the previous works in the literature where the control acts at the boundary,
the control for the entire system acts at an inside point of the PDE domain
whereas the PDE acts in the linear ODE by a Neumann connection. We use the
infinite dimensional backstepping design to convert system under consideration
to an exponentially target system. By invertibility of the design and Lyapunov
analysis, we prove the well posedness and exponential stability of such system.

Analysis of PDEs