Exponential stability of a coupled Heat-ODE system

This paper addresses the feedback stabilization of an interconnected Heat-ODE system with the Dirichlet interconnection. The “interconnection” between the heat equation and the ODE are bi-directional, in which the Dirichlet boundary observation of the heat equation is fed into the ODE, and the velocity of ODE is flowed into the boundary of the heat equation. The semigroup approach is adopted in investigation. By a detailed analysis, we obtain the asymptotic expressions of eigenvalues and eigenfunctions. It is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the Co-semigroup, and as a consequence, the exponential stability of the system is then followed. Numerical simulations are presented.