Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

Abstract

We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions M ± for one-parametric families of finitedimensional 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.Partly supported by MINECO/FEDER (Spain) under project MTM2015-66330-P and by European Commission under project H2020-MSCA- ITN-2014. where x ∈ R n and u ∈ R m , together with the quadratic form (supply rate)The functions A 0 , B 0 , G 0 , g 0 , and R 0 are assumed to be bounded and uniformly continuous functions on R, with values in the sets of real matrices of the appropriate dimensions. In addition, G and R are symmetric, and R(t) ≥ ρI m for a common ρ > 0 and all t ∈ R. We also fix x 0 ∈ R n and introduce the quadratic functionalthose for which u belongs to L 2 ((0, ∞), R m ) and the solution x(t) of (1.1) for this control with x(0) = x 0 belongs to L 2 ((0, ∞), R n ). The problem to consider is that of minimizing J x0 relative to the set of admissible pairs. By means of a standard construction (the so called hull or Bebutov construction, which we will summarize in Section 2), this problem can be included in a family, given by the control problemsand by the functionalsfor ω ∈ Ω and x 0 ∈ R n . Here, Ω is a compact metric space admitting a continuous flow, ω·t is the orbit of a point ω ∈ Ω, A, B, G, g, and R are bounded and uniformly continuous matrix-valued functions on Ω, G and R are symmetric, and R > 0. It is important to point out that Ω is minimal in the case of recurrence of the initial coefficients, which includes the autonomous, periodic, quasi-periodic, almost-periodic and almost-automorphic cases. The Pontryagin Maximum Principle relates the problem of minimizing J x0,ω to the properties of the family of linear Hamiltonian systemswhere z = [ x y ] for x, y ∈