Stabilization Of A First-order Hyperbolic System With An Intermediate Point And A Spatial Variable Coefficient

In this paper,we consider the stabilization of a first-order hyperbolic system with special spatial variable coefficient(exponential type)and intermediate point by Backstepping method.Our objective is to achieve the exponential stabilization of the system by designing suitable controller.Firstly,we choose an effective Backstepping transformation to transform the original system into a target system with exponential stability,meanwhile,we derive the kernel function.Secondly,we calculate the exact solution of the kernel function.Finally,according to the boundedness of the transformation and the stability of the target system,we obtain a closed-loop system with exponential stability.Furthermore,we extending the spatial variable coefficient of exponential type to the spatial variable coefficient of general derivable function,and also use Backstepping method to discuss the stability and the exact controllability by boundary control.It is worth mentioning that we can not calculate the exact solution of the kernel function at this time,but can prove the existence of the kernel function by using successive approximation method.Following the similar steps,we obtain the feedback controller and prove the exact controllability of the system.For the stabilization of the first-order hyperbolic system,some results consider the first-order hyperbolic system with an intermediate point and constant coefficient.Different from the existing results,this paper considers a first-order hyperbolic equation with spatial variable coefficient and intermediate point.Finally,simulation results and special condition show the validity of the designed controller.