2,923 research outputs found
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
Dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems
We derive a necessary and sufficient condition of linear dynamical stability
for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF)
model. The condition is expressed by an explicit disequality that has to be
satisfied by the stationary state, and it generalizes the known disequality for
homogeneous stationary states. In addition, we derive analogous disequalities
that express necessary and sufficient conditions of formal stability for the
stationary states. Their usefulness, from the point of view of linear dynamical
stability, is that they are simpler, although they provide only sufficient
criteria of linear stability. We show that for homogeneous stationary states
the relations become equal, and therefore linear dynamical stability and formal
stability become equivalent.Comment: Submitted to Journal of Statistical Mechanics: Theory and Experimen
Weakly nonlocal nonequilibrium thermodynamics: the Cahn-Hilliard equation
The Cahn-Hilliard and Ginzburg-Landau (Allen-Cahn) equations are derived from
the second law. The intuitive approach by separation of full divergences is
supported by a more rigorous method, based on Liu procedure and a constitutive
entropy flux. Thermodynamic considerations eliminate the necessity of
variational techniques and explain the role of functional derivatives.Comment: 13 page
Optimization Algorithms as Robust Feedback Controllers
Mathematical optimization is one of the cornerstones of modern engineering
research and practice. Yet, throughout all application domains, mathematical
optimization is, for the most part, considered to be a numerical discipline.
Optimization problems are formulated to be solved numerically with specific
algorithms running on microprocessors. An emerging alternative is to view
optimization algorithms as dynamical systems. Besides being insightful in
itself, this perspective liberates optimization methods from specific numerical
and algorithmic aspects and opens up new possibilities to endow complex
real-world systems with sophisticated self-optimizing behavior. Towards this
goal, it is necessary to understand how numerical optimization algorithms can
be converted into feedback controllers to enable robust "closed-loop
optimization". In this article, we focus on recent control designs under the
name of "feedback-based optimization" which implement optimization algorithms
directly in closed loop with physical systems. In addition to a brief overview
of selected continuous-time dynamical systems for optimization, our particular
emphasis in this survey lies on closed-loop stability as well as the robust
enforcement of physical and operational constraints in closed-loop
implementations. To bypass accessing partial model information of physical
systems, we further elaborate on fully data-driven and model-free operations.
We highlight an emerging application in autonomous reserve dispatch in power
systems, where the theory has transitioned to practice by now. We also provide
short expository reviews of pioneering applications in communication networks
and electricity grids, as well as related research streams, including extremum
seeking and pertinent methods from model predictive and process control, to
facilitate high-level comparisons with the main topic of this survey
3-D Velocity Regulation for Nonholonomic Source Seeking Without Position Measurement
We consider a three-dimensional problem of steering a nonholonomic vehicle to
seek an unknown source of a spatially distributed signal field without any
position measurement. In the literature, there exists an extremum seeking-based
strategy under a constant forward velocity and tunable pitch and yaw
velocities. Obviously, the vehicle with a constant forward velocity may exhibit
certain overshoots in the seeking process and can not slow down even it
approaches the source. To resolve this undesired behavior, this paper proposes
a regulation strategy for the forward velocity along with the pitch and yaw
velocities. Under such a strategy, the vehicle slows down near the source and
stays within a small area as if it comes to a full stop, and controllers for
angular velocities become succinct. We prove the local exponential convergence
via the averaging technique. Finally, the theoretical results are illustrated
with simulations.Comment: submitted to IEEE TCST;12 pages, 10 figure
- …