We analyze the presence of exponential dichotomy (ED) and of global existence of
Weyl functions M± for one-parametric families of finite-dimensional nonautonomous linear
Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed
from an initial one in a direction which does not satisfy the classical Atkinson condition:
either they do not have ED for any value of the parameter; or they have it for at least all the
nonreal values, in which case theWeyl functions exist and are Herglotz. When the parameter
varies in the real line, and if the unperturbed family satisfies the properties of exponential
dichotomy and global existence of M+, then these two properties persist in a neighborhood
of 0 which agrees either with the whole real line or with an open negative half-line; and in
this last case, the ED fails at the right end value. The properties of ED and of global existence
of M+ are fundamental to guarantee the solvability of classical minimization problems given
by linear–quadratic control processes.
