In this paper we open a new direction in the study of principal solutions for nonoscillatory linear Hamiltonian systems. In the absence of the controllability assumption, we introduce the minimal principal solution at infinity, which is a generalization of the classical principal solution (sometimes called the recessive solution) for controllable systems introduced by W.T.Reid, P.Hartman, and/or W.A.Coppel. The term ``minimal'' refers to the rank of the solution. We show that the minimal principal solution is unique (up to a right nonsingular multiple) and state its basic properties. We also illustrate our new theory by several examples.

Last change: November 28, 2013. (c) Roman Simon Hilscher