Stability Analysis Of Solutions Of A Class Of Interval Differential Equation

In the process of differential equation modeling,due to the complexity of practical problems or the limitations of people’s understanding,many uncertain parameters can only provide a certain range of interval changes,so the interval differential equation comes into being.Currently,the research on the properties of the solutions of interval differential equations is still in the exploratory stage,especially the research results on stability are relatively small.The main difficulty is that the interval space under standard interval arithmetic is not linear space.To overcome difficulties,this thesis uses constrained interval arithmetic to study the stability of solutions of a class of interval differential equations.By using constrained interval arithmetic,the interval uncertainty can be transformed into parameter uncertainty,and the interval space where the problem is located can be transformed into real number space(parameter space),so as to obtain stability results similar to those of classical ordinary differential equations.This similarity has great advantages in dealing with systems with interval uncertainty.The main work and results are as follows:(1)The matrix theory is extended to the constrained interval environment,including the eigenvalue of the interval matrix,the exponential function of the interval matrix,and the Jordan standardization of the interval matrix,which to some extent enriches constrained interval analysis.(2)Based on the stability of equilibrium points of interval two-dimensional linear differential equations,the stability of equilibrium points of interval two-dimensional semi-linear differential equations is studied by constructing Lyapunov functions based on constrained interval arithmetic.It is proved that when the perturbation term satisfies certain conditions,the type of equilibrium points of the equation is still determined by its linear part.(3)Combining constrained interval theory and classical Lyapunov stability theory,several definitions of the stability of zero solutions of interval n-dimensional differential equations are given.Based on the interval matrix theory,using the constrained interval representation of the fundamental solution matrix and the constrained interval results of the eigenvalue problem,the necessary and sufficient conditions for the asymptotic stability,stability and instability of the zero solution of the interval n-dimensional linear differential equations are obtained.Using Gronwall inequality to estimate the constrained interval solutions of interval n-dimensional semi-linear differential equations,a conclusion is obtained that the zero solution of interval ndimensional semi-linear differential equations is asymptotically stable under certain conditions.