The Research For An Unstable Differential Equation With Time Delay

In this paper, we researched a series of unstable differential equation that subject to time delay. There are three main parts in this paper. The first part is a research about a problem of output feedback stabilization of an unstable wave equation with observations subject to time delay. The second part is a research of a problem of an energy conservation and emission reduction system with feedback of time delay. The third part is a research about stabilization of a system of energy supply-demand under government regulation with time delay.In the first part, we researched the problem of output feedback stabilization of an unstable wave equation with observations subject to time delay. Firstly, we proved the observability of output, based which we designed observation of system and proved feasibility of it. We also designed predictor of system because of the exit of time delay and proved feasibility of it too. Secondly, using the method of "Backstepping" we solved the unstable boundary condition and obtained form of control that can stabilize the original system. Finally, using the state of observation and predictor to form control and using the control to the original system, we proved stability of system. Numerical simulation verified theories.In the second part, we researched the stability of system of energy conservation and emission reduction under time-delayed feedback control. Firstly, we established the model of energy conservation and emission under time-delayed feedback control. Secondly, we analyzed stability of equilibrium points and properties of Hopf-bifurcation. Finally, numerical simulation verified theories.In the third part, we researched a system of energy supply-demand and government regulation. Firstly, we established the model of energy supply-demand system without time delay and analyzed the stability of it. Secondly, we established the model of energy supply-demand system with time delay and analyzed the stabilization and properties of Hopf-bifurcation respectively. Finally, numerical simulation verified theories.The innovations in this paper. First, we resolved the problem of stabilization in unstable wave equation with observations subject to time delay under condition of uncoordinated control. Second, we used the feedback of time delay to control the system of energy conservation and emission reduction. Second, we established a new system of energy supply-demand and government regulation and analyzed the influence of time delay to the system. The difficulties in this paper are the prove of observability inequation, the design of "Backstepping" in unstable boundary condition and the analysis of time delay in conditions of stability switch.