  A key issue in dimension reduction of dissipative dynamical systems with
spectral gaps is the identification of slow invariant manifolds. We present
theoretical and numerical results for a variational approach to the problem of
computing such manifolds for kinetic models using trajectory optimization. The
corresponding objective functional reflects a variational principle that
characterizes trajectories on, respectively near, slow invariant manifolds. For
a two-dimensional linear system and a common nonlinear test problem we show
analytically that the variational approach asymptotically identifies the exact
slow invariant manifold in the limit of both an infinite time horizon of the
variational problem with fixed spectral gap and infinite spectral gap with a
fixed finite time horizon. Numerical results for the linear and nonlinear model
problems as well as a more realistic higher-dimensional chemical reaction
mechanism are presented.
