Noncollocated feedback stabilization of a heat conduction process on a nonuniform ring

In this article, we study the stabilization of a 1-d heat conduction process on a nonuniform ring, where the heat flux at $z=1$ is fed back to $z=0$ through a recycle loop, and two noncollocated point temperatures at $z=1$ and $z_0\in (0,1)$ are available to be measured. The instability of the heat system comes from two parts: one is the boundary recycle, and the other is the distributed terms of the heat equation. The static output control is designed at the left boundary $z=0$ to overcome the instability, and the admissible value ranges of feedback gains are concluded by spectral analysis so that the closed-loop system is shown to be well-posed and exponentially stable. The numerical simulations are carried out to demonstrate the effectiveness of the proposed controller.