Stability Analysis Of Time-Delay ODE/PDE Systems

The models with time-delay can describe the motion law and dynamic behavior of each system more comprehensively and accurately.All along,it has drawn the close attention of experts and scholars in various fields,and it has been widely studied and explored.The research content of this paper can be divided into two aspects.On the one hand,we study the design of boundary delay feedback controller and exponential stabilization of a class of simplified Saint-Venant equations.On the other hand,the stability of a ring neural network system with nine time-delays and ten parameters is discussed.In chapter 1,the research background,research status and significance of the stability of open-channel system and ring neural network system with time-delay are discussed,respectively.The simplified prisms open channel model with constant slope,rectangular section and no side flow and the ring neural network model with time delay are emphatically introduced.In chapter 2,a boundary feedback controller in the form of PDP(linear combination of instantaneous position and time-delay position)is designed for a single open-channel system described by a one-dimensional hyperbolic PDE model.Then,the semigroup theory is used to prove that the closed-loop system satisfies the well-posedness of the solution under certain conditions.Then,the sufficient conditions to guarantee that the solution of the closed-loop system decreases exponentially on the L~2-norm are obtained by using the constructed Lyapunov candidate functions.Finally,the asymptotic spectrum of the coupled system is analyzed,and the eigenvalues and eigenfunctions of the system are given,respectively.In Chapter 3,we extend the conclusion of a single open-channel system in Chapter 2 to star-shaped open-channel network with n subsystems.In the same way,by constructing weighted Lyapunov function,we discuss the exponential stability of the solution of the closed-loop system on the L~2-norm.Sufficient conditions for control parameters and time delay are given,and it is proved that the online output is regulated to a specified reference signal.In chapter 4,we study a class of four-element bidirectional ring neural network system described by ODE model with multiple delays and parameters.Firstly,the characteristic equation of the system is obtained by simple linearization near the trivial solution.Then,it is factored and converted into the seriation of four first-order exponential polynomials.Moreover,according to the characteristics of zero distribution of exponential polynomial,the stability criterions of the system are given,which are delay-dependent and delay-independent.Finally,a detailed comparison and analysis between the obtained results and previous research results are conducted,and the numerical simulation is combined to indicate the universality and superiority of the theoretical analysis.