| In this paper,we are concerned with the prescribed-time stabilization of unsta-ble parabolic PDEs with disturbance,where the prescribed timeis independent of the initial state of the system.By means of the method of active disturbance rejection control（ADRC）,the estimation and the compensation of disturbances are carried out by using backstepping transform during the feedback process,where the backstepping transform is combined with time-varying kerner function.And express the time-varying kerner function by using the Bessel function.Finally,the system is stabilized within a prescribed time.Firstly,the disturbance estimator of the controlled system is constructed,and the stability of the equivalent system of the estimator is studied under transformations such as backstepping with time-varying kernel function.Then,the stability of the disturbance estimator is analyzed and the disturbance and state are estimated within a prescribed time.Next,an output feedback controller is designed.Through the help of backstepping transformation,the equivalent system of the controlled system is analyzed,which is similar to the previous section.Thus,the output feedback control of the controlled system is obtained,and the closed-loop system is also determined.Finally,the stability of the closed-loop system is proved.By using the Lyapunov function to prove the stability of the equivalent closed-loop system within a prescribed time,the estimation of the solution of the time-varying kernel function equation is used during the variable estimation.Based on the invertible transformations,the stability of the closed-loop system is proved within the prescribed time,i.e.the system state ‖μ（·,t）‖L2（0,1）→0 as t→T.It is worth noting that in this paper due to the complexity of the system,the properties of equivalent systems are often studied first,in which the equivalent trans-formation of the system cannot be separated from the use of the backstepping trans-formation with the time-varying kernel function. |
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