1,899 research outputs found
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
Target Patterns in a 2-D Array of Oscillators with Nonlocal Coupling
We analyze the effect of adding a weak, localized, inhomogeneity to a two
dimensional array of oscillators with nonlocal coupling. We propose and also
justify a model for the phase dynamics in this system. Our model is a
generalization of a viscous eikonal equation that is known to describe the
phase modulation of traveling waves in reaction-diffusion systems. We show the
existence of a branch of target pattern solutions that bifurcates from the
spatially homogeneous state when , the strength of the
inhomogeneity, is nonzero and we also show that these target patterns have an
asymptotic wavenumber that is small beyond all orders in .
The strategy of our proof is to pose a good ansatz for an approximate form of
the solution and use the implicit function theorem to prove the existence of a
solution in its vicinity. The analysis presents two challenges. First, the
linearization about the homogeneous state is a convolution operator of
diffusive type and hence not invertible on the usual Sobolev spaces. Second, a
regular perturbation expansion in does not provide a good ansatz
for applying the implicit function theorem since the nonlinearities play a
major role in determining the relevant approximation, which also needs to be
"correct" to all orders in . We overcome these two points by
proving Fredholm properties for the linearization in appropriate Kondratiev
spaces and using a refined ansatz for the approximate solution, which obtained
using matched asymptotics.Comment: 39 pages, 1 figur
The Quark Propagator from the Dyson-Schwinger Equations: I. the Chiral Solution
Within the framework of the Dyson-Schwinger equations in the axial gauge, we
study the effect that non-perturbative glue has on the quark propagator. We
show that Ward-Takahashi identities, combined with the requirement of matching
perturbative QCD at high momentum transfer, guarantee the multiplicative
renormalisability of the answer. Technically, the matching with perturbation
theory is accomplished by the introduction of a transverse part to the
quark-gluon vertex. We show that this transverse vertex is crucial for chiral
symmetry breaking, and that massless solutions exist below a critical value of
the strong coupling constant. Using the gluon propagator that we previously
calculated, we obtain small corrections to the quark propagator, which keeps a
pole at the origin in the chiral phase.Comment: 21 pages, 6 figures; McGill/94-24, SHEP 93/94-26 We generalise our
results by showing that they are not sensitive to the specific choice that we
make for the transverse vertex. We illustrate that fact in two new figure
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