Linearization For Some Classes Of Impulsive Differential Equations

Since 1960,classical Hartman-Grobman theorem is extensively studied in different directions.In 1990,Fenner and Pinto[36]first introduced Hartman’s linearization prob-lem for a class of ODEs with impulse effect.Based on exponential dichotomy,Bellman inequality and Banach fixed point theory,we present two versions of Hartman-Grobman theorem for two kinds of impulsive system.Some new results are obtained.This paper has three chapters:In chapter 1,we briefly introduce the research background.Some definitions and lemmas are also introduced in this chapter.In chapter 2,we give a new version the Hartman-Grobman for the autonomous impulsive system when the nonlinear term is unbounded.Reinfelds[40]considered the globally strong dynamical equivalence between nonlinear impulsive system and its reduced system.Different from consideration from Reinfelds[40],we considered the topological equivalence between the following impulsive nonlinear and linear systems.Namely,we prove the nonlinear impulsive system is topologically conjugated to the linear system where ‖g(x1,x2)‖≤γ‖x1‖ +μ,‖g(x1,x2)‖≤γ‖x1‖ + μ,that is,the nonlinear term is unbounded.In chapter 3,we discuss the topological equivalence of a class of nonautonomous impulsive system with special structures.In Xia et al[37],the authors improved the lin-earization theorem under the condition that the impulsive linear system partially satisfies Is condition.Moveover,for the equivalent function H(t,x),they proved that the equiva-lent function H(t,x)is always Holder continuous.In this chapter,we proved that a class of nonautonomous impulsive system with special structures is topologically conjugated to its linear part.Moreover,for the equivalent functions H(t,x),G(t,y)(G(t,·)= H-1(t,·)),we prove that 1° for all t,we have‖H(t,x1)-H(t,x2)‖ ≤ p‖x1-x2‖(constant p>0).2° for all t,we have‖G(t,y1)-G(t,y2)‖≤q‖≤y1-y2‖(constant q>0).