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

Let S be an operator in a Banach space H and S ⁱ (u) (i = 0, 1, ..., u ∈ H) be the evolutionary process specified by S. The following problem is considered: for a given point z ₀ and a given initial condition a ₀, find a correction l such that the trajectory {S ⁱ (a ₀ + l)} approaches }S ⁱ (z ₀)} for 0 < i ≤ n. This problem is reduced to projecting a ₀ on the manifold ?^?(z ₀, f ⁽ⁿ⁾) defined in a neighborhood of z ₀ and specified by a certain function f ⁽ⁿ⁾. In this paper, an iterative method is proposed for the construction of the desired correction u = a ₀ + 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 ?^?(z ₀, f) in a neighborhood of a trajectory of hyperbolic type is one of the possible applications of the proposed method. For the points in ?^?(z ₀, f), the value of n can be chosen arbitrarily large.