Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 03, pp. 1-26.

Title: Internal stabilization of interconnected heat-wave equations
       
Authors: Xiu-Fang Yu  (Beijing Institute of Technology, Beijing, China)
         Jun-Min Wang (Beijing Institute of Technology, Beijing, China)
         Han-Wen Zhang (Shanxi Univ., Taiyuan Shanxi, China)
         
Abstract: 
This article concerns the internal stabilization problem of 1-D interconnected heat-wave
equations, where information exchange and the two actuators occur at the adjacent
side of the two equations. By designing an inverse back-stepping transformation,
the original system is converted into a dissipative target system.
Moreover, we investigate the eigenvalues distribution and the corresponding
eigenfunctions of the closed-loop system by an asymptotic analysis method.
This shows that the spectrum of the system can be divided into two families:
one distributed along the a line parallel to the left side of the imaginary axis
and symmetric to the real axis, and the other on the left half real axis.
Then we work on the properties of the resolvent operator and we verify that
the root subspace is complete.
Finally, we prove that the closed-loop system is exponentially stable. 

Submitted September 2, 2022. Published January 06, 2023.
Math Subject Classifications: 93C20, 93D15.
Key Words: Heat-wave equation; boundary control; spectrum; root subspace; exponential stability.