In this report, I shall firstly present the background of uniform exponential stability approximating of elastic vibration systems and the existed science problem and challenge difficulty. Secondly, I introduce several methods, which are newly designed by collaborators and me, to overcome difficulties and restore the exponential stability approximating of the original elastic vibration systems. Thirdly, I provide several important applications of uniform exponential stability in the state reconstruct and numerical approximating of control. At last, I draw some conclusions and give some further research directions.