The method of asymptotic stabilization to a given trajectory based on a correction of the initial data

Abstract

Let S be an operator in a Banach space H and Si(u) (i = 0, 1, ..., u ∈ H) be the evolutionary process specified by S. The following problem is considered: for a given point z0 and a given initial condition a0, find a correction l such that the trajectory {Si(a0 + l)} approaches }Si(z0)} for 0 < i ≤ n. This problem is reduced to projecting a0 on the manifold ℳ−(z0, f(n)) defined in a neighborhood of z0 and specified by a certain function f(n). In this paper, an iterative method is proposed for the construction of the desired correction u = a0 + l. The convergence of the method is substantiated, and its efficiency for the blow-up Chafee-Infante equation is verified. A constructive proof of the existence of a locally stable manifold ℳ−(z0, f) in a neighborhood of a trajectory of hyperbolic type is one of the possible applications of the proposed method. For the points in ℳ−(z0, f), the value of n can be chosen arbitrarily large.