Principal solutions at infinity for time scale symplectic systems without controllability condition

In this paper we introduce a new concept of a principal solution at infinity for nonoscillatory symplectic dynamic systems on time scales. The main ingredient is that we avoid the controllability (or normality) condition, which is traditionally assumed in this theory in the current literature. We show that the principal solutions at infinity can be classified according to the eventual rank of their first component and that the principal solutions exist for all values of the rank between explicitly given minimal and maximal values. The minimal value of the rank is connected with the eventual order of abnormality of the system and it gives rise to the so-called minimal principal solution at infinity. We show that the uniqueness property of the principal solutions at infinity is satisfied only by the minimal principal solution. In this study we unify and extend to arbitrary time scales the recently introduced theory of principal and recessive solutions at infinity for possibly abnormal (continuous time) linear Hamiltonian differential systems and (discrete time) symplectic systems. Moreover, the new theory on time scales also shows that in some results from the continuous time theory the needed assumptions can be simplified.