The exponential stability of an unstable ODE with compensation of a heat equation through Neumann interconnections

This paper addresses the feedback stabilization of a coupled heat-ODE system through the Neumann boundary interconnections, where the boundary heat flux vx(1, t) is fed into the ODE, while the velocity feedback of ODE is flowed into the boundary of heat equation, so a direct bi-directional feedback between the ODE and the heat equation is established. It is found that the dissipative damping is produced in the heat equation via the boundary connections only, and then the heat equation is considered as the controller of the whole system. Based on the semigroup approach and Riesz basis method, the well-posedess and exponential stability of the system are deduced. Finally, some numerical simulations are presented to show the differences and merits between delay compensator, heat PDE compensator via Dirichlet interconnections, and heat PDE compensator via Neumann interconnections.