111,326 research outputs found
Time-dependent mode structure for Lyapunov vectors as a collective movement in quasi-one-dimensional systems
Time dependent mode structure for the Lyapunov vectors associated with the
stepwise structure of the Lyapunov spectra and its relation to the momentum
auto-correlation function are discussed in quasi-one-dimensional many-hard-disk
systems. We demonstrate mode structures (Lyapunov modes) for all components of
the Lyapunov vectors, which include the longitudinal and transverse components
of their spatial and momentum parts, and their phase relations are specified.
These mode structures are suggested from the form of the Lyapunov vectors
corresponding to the zero-Lyapunov exponents. Spatial node structures of these
modes are explained by the reflection properties of the hard-walls used in the
models. Our main interest is the time-oscillating behavior of Lyapunov modes.
It is shown that the largest time-oscillating period of the Lyapunov modes is
twice as long as the time-oscillating period of the longitudinal momentum
auto-correlation function. This relation is satisfied irrespective of the
particle number and boundary conditions. A simple explanation for this relation
is given based on the form of the Lyapunov vector.Comment: 39 pages, 21 figures, Manuscript including the figures of better
quality is available from http://www.phys.unsw.edu.au/~gary/Research.htm
Economic MPC of Nonlinear Systems with Non-Monotonic Lyapunov Functions and Its Application to HVAC Control
This paper proposes a Lyapunov-based economic MPC scheme for nonlinear sytems
with non-monotonic Lyapunov functions. Relaxed Lyapunov-based constraints are
used in the MPC formulation to improve the economic performance. These
constraints will enforce a Lyapunov decrease after every few steps. Recursive
feasibility and asymptotical convergence to the steady state can be achieved
using Lyapunov-like stability analysis. The proposed economic MPC can be
applied to minimize energy consumption in HVAC control of commercial buildings.
The Lyapunov-based constraints in the online MPC problem enable the tracking of
the desired set-point temperature. The performance is demonstrated by a virtual
building composed of two adjacent zones
Constructions of Strict Lyapunov Functions for Discrete Time and Hybrid Time-Varying Systems
We provide explicit closed form expressions for strict Lyapunov functions for
time-varying discrete time systems. Our Lyapunov functions are expressed in
terms of known nonstrict Lyapunov functions for the dynamics and finite sums of
persistency of excitation parameters. This provides a discrete time analog of
our previous continuous time Lyapunov function constructions. We also construct
explicit strict Lyapunov functions for systems satisfying nonstrict discrete
time analogs of the conditions from Matrosov's Theorem. We use our methods to
build strict Lyapunov functions for time-varying hybrid systems that contain
mixtures of continuous and discrete time evolutions.Comment: 14 pages. Accepted for publication in Nonlinear Analysis: Hybrid
Systems and Applications on September 6, 200
Continuity Results and Estimates for the Lyapunov Exponent of Brownian Motion in Random Potential
We collect some applications of the variational formula established by
Schr\"oder (1988) and Rue\ss (2013) for the quenched Lyapunov exponent of
Brownian motion in stationary and ergodic nonnegative potential. We show for
example that the Lyapunov exponent for nondeterministic potential is strictly
lower than the Lyapunov exponent for the averaged potential. The behaviour of
the Lyapunov exponent under independent perturbations of the underlying
potential is examined. And with the help of counterexamples we are able to give
a detailed picture of the continuity properties of the Lyapunov exponent.Comment: 20 pages, 1 figure, some references update
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