脉冲系统中闭集稳定性与集值映射连续性的关系

Relationships Between Stability of Closed Sets and Continuity of Set-valued Maps in Impulsive Dynamical Systems

为进一步探讨脉冲动力系统中闭集的局部稳定性和集值映射的连续性的关系,在脉冲动力系统中引入闭集在某一点处稳定的定义及其等价条件,参照一般连续动力系统中的情形,以度量空间中的HAUSDORFF度量为工具,讨论脉冲动力系统中两类特殊的闭集,即正延伸集和正延伸极限集的稳定性与相应的集值映射的连续性之间的关系。研究表明:集值映射D^(+)(或K^(+))在x点上半连续当且仅当集合D^(+)(x)(或K^(+)(x))在x点稳定,映射L^(+)在x点上半连续当且仅当闭集L^(+)(x)在x点最终稳定,映射J^(+)在x点上半连续,蕴含着闭集J^(+)(x)在x点最终稳定,反之,若J^(+)(x)在x点一致最终稳定,那么映射J^(+)在x点上半连续。

In this paper,we deal with the relationship between the local stability of closed sets and the continuity of set-valued maps in impulsive dynamical systems.We introduce the definition of stability for closed sets at a point and its equivalent conditions,and investigate by HAUSDORFF metric the relationships between the stability of two special closed sets—positive prolongations and positive prolongational limit sets—and the continuity of corresponding set-valued maps,as in the case of continuous dynamical systems.It shows that the map D^(+)(orK^(+))is upper semi-continuous at x if and only if closed set D^(+)(x)(orK^(+)(x))is stable at x,the map L^(+)is upper semi-continuous at x if and only if closed set L^(+)(x)is eventually stable at x,the map J^(+)is upper semi-continuous at x implies L^(+)(x)is eventually stable at x;and conversely,if L^(+)(x)has uniform eventual stability at x,then J^(+)is upper semi-continuous at x.