A method for the asymptotic stabilization to a given trajectory based on the initial data

Let $S$ be an operator in a Banach space $H$ and $S^i(u)$, $i=0,1,\dots,u\in H$ 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 $\mathscr M^-(z_0,f^{(n)})$ 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 $\mathscr M^-(z_0,f)$ in a neighborhood of a trajectory of hyperbolic type is one of the possible applications of the proposed method. For the points in $\mathscr M^-(z_0,f)$, the value of $n$ can be chosen arbitrarily large.