4,966 research outputs found
Small gain theorems for large scale systems and construction of ISS Lyapunov functions
We consider interconnections of n nonlinear subsystems in the input-to-state
stability (ISS) framework. For each subsystem an ISS Lyapunov function is given
that treats the other subsystems as independent inputs. A gain matrix is used
to encode the mutual dependencies of the systems in the network. Under a small
gain assumption on the monotone operator induced by the gain matrix, a locally
Lipschitz continuous ISS Lyapunov function is obtained constructively for the
entire network by appropriately scaling the individual Lyapunov functions for
the subsystems. The results are obtained in a general formulation of ISS, the
cases of summation, maximization and separation with respect to external gains
are obtained as corollaries.Comment: provisionally accepted by SIAM Journal on Control and Optimizatio
Numerical Construction of LISS Lyapunov Functions under a Small Gain Condition
In the stability analysis of large-scale interconnected systems it is
frequently desirable to be able to determine a decay point of the gain
operator, i.e., a point whose image under the monotone operator is strictly
smaller than the point itself. The set of such decay points plays a crucial
role in checking, in a semi-global fashion, the local input-to-state stability
of an interconnected system and in the numerical construction of a LISS
Lyapunov function. We provide a homotopy algorithm that computes a decay point
of a monotone op- erator. For this purpose we use a fixed point algorithm and
provide a function whose fixed points correspond to decay points of the
monotone operator. The advantage to an earlier algorithm is demonstrated.
Furthermore an example is given which shows how to analyze a given perturbed
interconnected system.Comment: 30 pages, 7 figures, 4 table
Geometric Properties of Isostables and Basins of Attraction of Monotone Systems
In this paper, we study geometric properties of basins of attraction of
monotone systems. Our results are based on a combination of monotone systems
theory and spectral operator theory. We exploit the framework of the Koopman
operator, which provides a linear infinite-dimensional description of nonlinear
dynamical systems and spectral operator-theoretic notions such as eigenvalues
and eigenfunctions. The sublevel sets of the dominant eigenfunction form a
family of nested forward-invariant sets and the basin of attraction is the
largest of these sets. The boundaries of these sets, called isostables, allow
studying temporal properties of the system. Our first observation is that the
dominant eigenfunction is increasing in every variable in the case of monotone
systems. This is a strong geometric property which simplifies the computation
of isostables. We also show how variations in basins of attraction can be
bounded under parametric uncertainty in the vector field of monotone systems.
Finally, we study the properties of the parameter set for which a monotone
system is multistable. Our results are illustrated on several systems of two to
four dimensions.Comment: 12 pages, to appear in IEEE Transaction on Automatic Contro
On multichannel solutions of nonlinear Schr\"{o}dinger equations: algorithm, analysis and numerical explorations
We apply the method of modulation equations to numerically solve the NLS with
multichannel dynamics, given by a trapped localized state and radiation. This
approach employs the modulation theory of Soffer-Weinstein, which gives a
system of ODE's coupled to the radiation term, which is valid for all times. We
comment on the differences of this method from the well-known method of
collective coordinates.Comment: 16 pages, 9 figures. Added a comparison to collective coordinates
scheme that includes a radiation ter
A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS
Based on our previous work for solving the nonlinear Schrodinger equation
with multichannel dynamics that is given by a localized standing wave and
radiation, in this work we deal with the multichannel solution which consists
of a moving soliton and radiation. We apply the modulation theory to give a
system of ODEs coupled to the radiation term for describing the solution, which
is valid for all times. The modulation equations are solved accurately by the
proposed numerical method. The soliton and radiation are captured separately in
the computation, and they are solved on the translated domain that is moving
with them. Thus for a fixed finite physical domain in the lab frame, the
multichannel solution can pass through the boundary naturally, which can not be
done by imposing any existing boundary conditions. We comment on the
differences of this method from the collective coordinates.Comment: 19 pages, 7 figures. To appear on Phys. D. arXiv admin note: text
overlap with arXiv:1404.115
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
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