1,755 research outputs found
Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients
Recently, the problem of boundary stabilization for unstable linear
constant-coefficient coupled reaction-diffusion systems was solved by means of
the backstepping method. The extension of this result to systems with advection
terms and spatially-varying coefficients is challenging due to complex boundary
conditions that appear in the equations verified by the control kernels. In
this paper we address this issue by showing that these equations are
essentially equivalent to those verified by the control kernels for first-order
hyperbolic coupled systems, which were recently found to be well-posed. The
result therefore applies in this case, allowing us to prove H^1 stability for
the closed-loop system. It also shows an interesting connection between
backstepping kernels for coupled parabolic and hyperbolic problems.Comment: Submitted to IEEE Transactions on Automatic Contro
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
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
This work deals with the numerical solution of the monodomain and bidomain
models of electrical activity of myocardial tissue. The bidomain model is a
system consisting of a possibly degenerate parabolic PDE coupled with an
elliptic PDE for the transmembrane and extracellular potentials, respectively.
This system of two scalar PDEs is supplemented by a time-dependent ODE modeling
the evolution of the so-called gating variable. In the simpler sub-case of the
monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple
models for the membrane and ionic currents are considered, the
Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical
solutions of the bidomain and monodomain models exhibit wavefronts with steep
gradients, we propose a finite volume scheme enriched by a fully adaptive
multiresolution method, whose basic purpose is to concentrate computational
effort on zones of strong variation of the solution. Time adaptivity is
achieved by two alternative devices, namely locally varying time stepping and a
Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical
examples demonstrates thatthese methods are efficient and sufficiently accurate
to simulate the electrical activity in myocardial tissue with affordable
effort. In addition, an optimalthreshold for discarding non-significant
information in the multiresolution representation of the solution is derived,
and the numerical efficiency and accuracy of the method is measured in terms of
CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure
Unique continuation property with partial information for two-dimensional anisotropic elasticity systems
In this paper, we establish a novel unique continuation property for
two-dimensional anisotropic elasticity systems with partial information. More
precisely, given a homogeneous elasticity system in a domain, we investigate
the unique continuation by assuming only the vanishing of one component of the
solution in a subdomain. Using the corresponding Riemann function, we prove
that the solution vanishes in the whole domain provided that the other
component vanishes at one point up to its second derivatives. Further, we
construct several examples showing the possibility of further reducing the
additional information of the other component. This result possesses remarkable
significance in both theoretical and practical aspects because the required
data is almost halved for the unique determination of the whole solution.Comment: 14 pages, 1 figur
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