Stability Of A Class Of Systems Of Differential Equations With Variable Coefficients

In this paper, a class of delay differential equations with variable coefficients is studied. Article is divided into three parts:firstly, the output feedback stabilization of variable coefficients Euler-Bernoulli Beam Equations with time delay is considered; secondly, the output feedback stabilization of the variable coefficients one-dimensional wave equation with time delay is given; then, the stability of energy saving and emission reduction system with time delay is analyzed.The specific contents of the article are as follows. In the first part, the output feedback stability of the Euler-Bernoulli Beam Equations with variable coefficients under time delay is considered in this paper. By introducing a new variable well-posedness properties of the system have been given. Namely that, the solution exists and there is only one. And the state of the system is determined by the initial value of the system and the input, which indicates the validity of the system state and the observation value. Due to the delay it may not be observed in a certain period of time, the design of the observer and the predictor have been given. Then, based on the estimated state observations, we design a feedback controller and get the closed-loop system. By using Riesz base nature and equivalent transformation method to solve the differential equations with variable coefficients, and then we can estimate the error. Finally the state and the speed of the closed-loop system is simulated to verify the validity of the theory.In the second part, the feedback stabilization problem of one dimensional wave equation with variable coefficients under time delay is studied. By introducing a new variable the well-posedness of the system were given. Then the design of the observer and the predictor is given, and demonstrate the effectiveness of the two systems. A feedback controller is designed based on the observation, and it is proved that the closed loop system is exponential stability. Different from the previous one, this section considers the second order equation. In order to obtain the exponential decay of the disturbance term in the closed loop system, it is necessary to estimate the exponential decay. Finally the state and the speed of the closed-loop system is simulated to verify the validity of the theory.In the third part, a three dimensional differential equation model for energy saving and emission reduction systems is considered. Due to the persistence and lag of carbon emission, and the lag of economic growth information, the energy saving and emission reduction system model with time delay is established. By solving the equilibrium point of the model and analyzing the stability of the equilibrium point, the conditions of the equilibrium point asymptotically stable, unstable and the occurrence of Hopf bifurcation are obtained. In particular, the properties of Hopf bifurcation are given by the center manifold theory and the standard type method. Finally the numerical simulation shows the correctness of the theory.