A stability theory for perturbed differential equations

[摘要] The problem of determining the behavior of the solutions of a perturbeddifferential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered isX′=f(t,X)and the associated perturbed differential equation isY′=f(t,Y)+g(t,Y).The approach used is to examine the difference between the respective solutionsF(t,t0,x0)andG(t,t0,y0)of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the differenceG(t,t0,y0)−F(t,t0,x0)in the case where the initial valuesy0andx0are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.

[学科分类] 数学（综合）