229 research outputs found
Prescribed-Time Seeking of a Repulsive Source by Unicycle Angular Velocity Tuning
All the existing source seeking algorithms for unicycle models in GPS-denied
settings guarantee at best an exponential rate of convergence over an infinite
interval. Using the recently introduced time-varying feedback tools for
prescribed-time stabilization, we achieve source seeking in prescribed time,
i.e., the convergence to the source, without the measurements of the position
and velocity of the unicycle, in as short a time as the user desires, starting
from an arbitrary distance from the source. The convergence is established
using a singularly perturbed version of the Lie bracket averaging method,
combined with time dilation and time contraction operations. The algorithm is
robust, provably, even to an arbitrarily strong gradient-dependent repulsive
velocity drift emanating from the source
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Integrability Of A Singularly Perturbed Model Describing Gravity Water Waves On A Surface Of Finite Depth
Our work is closely connected with the problem of splitting of separatrices (breaking of homoclinic orbits) in a singularly perturbed model describing gravity water waves on a surface of finite depth. The singularly perturbed model is a family of singularly perturbed fourth-order nonlinear ordinary differential equations, parametrized by an external parameter (in addition to the small parameter of the perturbations). It is known that in general separatrices will not survive a singular perturbation. However, it was proven by Tovbis and Pelinovsky that there is a discrete set of exceptional values of the external parameter for which separatrices do survive the perturbation. Since our family of equations can be written in the Hamiltonian form, the question is whether or not survival of separatrices implies integrability of the corresponding equation. The complete integrability of the system is examined from two viewpoints: 1) the existence of a second first integral in involution (Liouville integrability), and 2) the existence of single-valued, meromorphic solutions (complex analytic integrability). In the latter case, a singular point analysis is done using the technique given by Ablowitz, Ramani, and Segur (the ARS algorithm) to determine whether the system is of Painlevé-type (P-type), lacking movable critical points. The system is shown by the algorithm to fail to be of P-type, a strong indication of nonintegrability
Expansion of a singularly perturbed equation with a two-scale converging convection term
In many physical contexts, evolution convection equations may present some
very large amplitude convective terms. As an example, in the context of
magnetic confinement fusion, the distribution function that describes the
plasma satisfies the Vlasov equation in which some terms are of the same order
as , being the characteristic gyrokinetic
period of the particles around the magnetic lines. In this paper, we aim to
present a model hierarchy for modeling the distribution function for any value
of by using some two-scale convergence tools. Following Fr\'enod \\&
Sonnendr\"ucker's recent work, we choose the framework of a singularly
perturbed convection equation where the convective terms admit either a high
amplitude part or a an oscillating part with high frequency . In this abstract framework, we derive an expansion with respect to the
small parameter and we recursively identify each term of this
expansion. Finally, we apply this new model hierarchy to the context of a
linear Vlasov equation in three physical contexts linked to the magnetic
confinement fusion and the evolution of charged particle beams
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