Stabilization of a second order ODE–heat system coupling at intermediate point

This paper considers the stabilization of a second order ODE–heat system coupling at an intermediate point under natural and checkable assumptions, which is motivated by the thermoelastic coupling physics arising in microelectromechanical systems (MEMS). A novel backstepping transformation form is proposed in this work and the feedback gain for the lumped-parameter component is constructed using the eigenvector of coefficient matrix in the ODE system. Then, we prove the existence of smooth kernels with second order continuously derivative for the forward and inverse transformations in the backstepping feedback control law design. At the same time, we show that the forward and inverse transformations are mutually invertible transformation pair. Finally, the effectiveness of the stabilization feedback controller design is shown with some numerical examples.