  We analyze the presence of exponential dichotomy (ED) and of global existence
of Weyl functions $M^\pm$ 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 the Weyl 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.
