58 research outputs found
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
A variational principle for computing slow invariant manifolds in dissipative dynamical systems
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.Comment: 16 pages, 5 figure
Minimal Curvature Trajectories: Riemannian Geometry Concepts for Model Reduction in Chemical Kinetics
In dissipative ordinary differential equation systems different time scales
cause anisotropic phase volume contraction along solution trajectories. Model
reduction methods exploit this for simplifying chemical kinetics via a time
scale separation into fast and slow modes. The aim is to approximate the system
dynamics with a dimension-reduced model after eliminating the fast modes by
enslaving them to the slow ones via computation of a slow attracting manifold.
We present a novel method for computing approximations of such manifolds using
trajectory-based optimization. We discuss Riemannian geometry concepts as a
basis for suitable optimization criteria characterizing trajectories near slow
attracting manifolds and thus provide insight into fundamental geometric
properties of multiple time scale chemical kinetics. The optimization criteria
correspond to a suitable mathematical formulation of "minimal relaxation" of
chemical forces along reaction trajectories under given constraints. We present
various geometrically motivated criteria and the results of their application
to three test case reaction mechanisms serving as examples. We demonstrate that
accurate numerical approximations of slow invariant manifolds can be obtained.Comment: 22 pages, 18 figure
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