Abstract: The effect of initial data on the state of nonlinear systems (evolution equation and ordinary differential equations) at a fixed later time T is an important subject to study. The maximum norm of disturbances at time T when the finite amplitude of initial data disturbances restricted in a region (called the initial disturbance region) is studied. By considering the solutions to the problems of backwards in time and the hypothesizing the continuously dependence of solutions to the initial dada, it is proved that if the maximum norm of disturbances at time T is attained, then the initial data are at the boundary of the initial disturbance region. The importance of the results is on reducing the computation labor of the maximum norm of the disturbances.

Key words: conditional optimal nonlinear perturbation, maximum principle, calculus of variation, conditional optimal nonlinear perturbation, maximum principle