5,185 research outputs found
Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus
We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional
torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small
data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain
BBM-like equations, including the equal width wave equation and the KdV-BBM
equation. Applications to the stabilization of the above equations are given.
In particular, we show that when an internal control acting on a moving
interval is applied in BBM equation, then a semiglobal exponential
stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove
that the BBM equation with a moving control is also locally exactly
controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H
s (T) for any s \geq 1
Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles
This paper studies the local exact controllability and the local
stabilization of the semilinear Schr\"odinger equation posed on a product of
intervals (). Both internal and boundary controls are considered,
and the results are given with periodic (resp. Dirichlet or Neumann) boundary
conditions. In the case of internal control, we obtain local controllability
results which are sharp as far as the localization of the control region and
the smoothness of the state space are concerned. It is also proved that for the
linear Schr\"odinger equation with Dirichlet control, the exact controllability
holds in whenever the control region contains a neighborhood
of a vertex
Second Repeating FRB 180814.J0422+73: Ten-year Fermi-LAT Upper Limits and Implications
The second repeating fast radio burst source, FRB 180814.J0422+73, was detected recently by the CHIME collaboration. We use the ten-year Fermi Large Area Telescope archival data to place a flux upper limit in the energy range of 100 MeV−10 GeV at the position of the source, which is ~1.1 × 10−11 erg cm−2 s−1 for a six-month time bin on average, and ~2.4 × 10−12 erg cm−2 s−1 for the entire ten-year time span. For the maximum redshift of z = 0.11, the ten-year upper limit of luminosity is ~7.3 × 1043 erg s−1. We utilize these upper limits to constrain the fast radio burst (FRB) progenitor and central engine. For the rotation-powered young magnetar model, the upper limits can pose constraints on the allowed parameter space for the initial rotational period and surface magnetic field of the magnetar. We also place significant constraints on the kinetic energy of a relativistic external shock wave, ruling out the possibility that there existed a gamma-ray burst (GRB) beaming toward Earth during the past ten years as the progenitor of the repeater. The case of an off-beam GRB is also constrained if the viewing angle is not much greater than the jet opening angle. All of these constraints are more stringent if FRB 180814.J0422+73 is at a closer distance
- …