Exponential Stabilization of a Heat-ODE System Cascaded at Two Intermediate Points

This paper considers the exponential stability of a heat-ODE system cascaded at two intermediate points. A new backstepping transformation with three kernels is introduced. Firstly, by adopting a reversible backstepping transformation to design a stabilizing feedback control law, we prove that the existence of smooth kernel functions in the forward and inverse transformations. Then, the exponential stability of the closed-loop system is proved based on the stability of target system and the invertibility of backstepping transformation. Finally, we verify the compatibility between the known assumption for the special case and the controllability assumption of this paper.