A stability theory for perturbed differential equations. (1979) - Gordon, Sheldon P.
Abstract : The problem of determining the behavior of the solutions of a perturbed differential 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, t 0, x 0 ) andG ( t, t 0, y 0 ) 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, t 0, y 0 ) − F ( t, t 0, x 0 ) in the case where the initial valuesy 0 andx 0 are 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.