Inertial manifolds via spatial averaging: a control-theoretic perspective
Mikhail Anikushin
TL;DR
The work develops a control-theoretic, operator-theoretic framework to study inertial-manifold construction via quadratic optimization in infinite-dimensional, possibly nonautonomous settings. By combining Lyapunov-Perron theory for exponential dichotomies with the geometry of Lagrangian subspaces, it proves the existence and uniform nonoscillation of stable Lagrangian bundles under both stationary frequency conditions and the Spatial Averaging Principle. It provides graph representations for these bundles, contraction-based solvability results, and continuity in spatial parameters, linking spectral conditions to controllability and Riccati-type structures. The results unify stationary and nonstationary quadratic optimization approaches, enabling uniform estimates and paving the way for inertial-manifold constructions beyond the classical spectral-gap paradigm in PDEs. This offers a robust toolkit for analyzing dissipative infinite-dimensional systems and their finite-dimensional reductions under nonautonomous forcing and averaging regimes.
Abstract
We develop a functional-analytical machinery for studying the quadratic regulator problem arising from spectra perturbations of infinite-dimensional dynamical systems. In particular, we are interested in applications to inertial manifolds theory. For certain nonautonomous Hamiltonian systems associated with such problems, we show the existence and uniform nonoscillation of stable Lagrangian bundles. This is done within the context of the classical frequency condition for stationary problems, as well as for nonstationary problems arising under the conditions of the Spatial Averaging Principle of J. Mallet-Paret and G.R. Sell.