Boundary Stabilization Of A Coupled Wave-ODE System With Internal Anti-damping

Boundary stabilization for a coupled system of a parabolic partial differential equation(PDE) and an ordinary differential equation(ODE) with internal anti-damping is discussed in this paper. Through backstepping transformation and state feedback design, exponential stabilization of the considered system is achieved.In this paper, the method of backstepping is used to stabilize a coupled wave-ODE system with internal anti-damping by means of decoupling them into a stable cascaded waveODE system. Firstly,this approach is the structure of the variable reversible transformation,which makes the system appear linear in the new variable, and then eliminate nonlinearity,exponential stability of the new system is obtained. But, before solving the kernel funtion of the backstepping transformation and state feedback equation for the system, one can eliminate the term ux(x, t) by introducing the new variable v(x, t) = e1/2xu(x, t). So the new inversion is made on the basis of the original anthor's existing results, the new inversion is carried out to solve the state feedback control.The existing results only contains the anti-damping term ?(x)ut(x, t), Whereas in the systems considered in this paper, in addition to the anti-damping term ?(x)ut(x, t), the term ux(x, t) is also added to this paper. So a new conversion is applied in this paper, then the design of the state feedback controller is carried out on the basis of the transformed system.Therefore, the results obtained in this paper not only further develop the established results,but also have important value in practice. So the main innovation of this paper is that the system is more complex, more practical and research value coupled systems considered in this paper.