Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control

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This monograph provides an in-depth analysis of the dynamics of linear Hamiltonian systems of general dimension with nonautonomous bounded and uniformly continuous coefficients, without initial assumptions on time-recurrence. It emphasizes the oscillation properties of solutions and presents a spectral theory suited for these systems. The authors extend well-known results applicable to autonomous or periodic coefficients, as well as in the nonautonomous two-dimensional case, while introducing substantial new theory even in these simpler contexts. Systematic use of Lagrange planes and symplectic matrices, alongside methods from topological dynamics and ergodic theory, is made throughout the work. Key analytical tools include Lyapunov exponents, Weyl matrices, exponential dichotomy, and weak disconjugacy, with the rotation number for linear Hamiltonian systems playing a central role. These concepts underpin the exploration of various themes related to linear-quadratic control problems, such as the linear regulator property, the Kalman-Bucy filter, infinite-horizon optimization, the nonautonomous Yakubovich Frequency Theorem, and Willems dissipativity. This book is a valuable resource for graduate students and researchers interested in nonautonomous differential equations, dynamical systems, ergodic theory, spectral theory of differential operators, and control theory.