2,786 research outputs found
Fast global null controllability for a viscous Burgers' equation despite the presence of a boundary layer
In this work, we are interested in the small time global null controllability
for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment
[0,1]. The second-hand side is a scalar control playing a role similar to that
of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two
controls (namely the interior one u(t) and the boundary one y(t,0)). In this
setting, we show that small time global null controllability still holds by
taking advantage of both hyperbolic and parabolic behaviors of our system. We
use the Cole-Hopf transform and Fourier series to derive precise estimates for
the creation and the dissipation of a boundary layer
An obstruction to small time local null controllability for a viscous Burgers' equation
In this work, we are interested in the small time local null controllability
for the viscous Burgers' equation on the line
segment , with null boundary conditions. The second-hand side is a
scalar control playing a role similar to that of a pressure. In this setting,
the classical Lie bracket necessary condition introduced by
Sussmann fails to conclude. However, using a quadratic expansion of our system,
we exhibit a second order obstruction to small time local null controllability.
This obstruction holds although the information propagation speed is infinite
for the Burgers equation. Our obstruction involves the weak norm of
the control . The proof requires the careful derivation of an integral
kernel operator and the estimation of residues by means of weakly singular
integral operator estimates
Throughput-Optimal Random Access with Order-Optimal Delay
In this paper, we consider CSMA policies for scheduling of multihop wireless
networks with one-hop traffic. The main contribution of this paper is to
propose Unlocking CSMA (U-CSMA) policy that enables to obtain high throughput
with low (average) packet delay for large wireless networks. In particular, the
delay under U-CSMA policy becomes order-optimal. For one-hop traffic, delay is
defined to be order-optimal if it is O(1), i.e., it stays bounded, as the
network-size increases to infinity. Using mean field theory techniques, we
analytically show that for torus (grid-like) interference topologies with
one-hop traffic, to achieve a network load of , the delay under U-CSMA
policy becomes as the network-size increases, and hence,
delay becomes order optimal. We conduct simulations for general random
geometric interference topologies under U-CSMA policy combined with congestion
control to maximize a network-wide utility. These simulations confirm that
order optimality holds, and that we can use U-CSMA policy jointly with
congestion control to operate close to the optimal utility with a low packet
delay in arbitrarily large random geometric topologies. To the best of our
knowledge, it is for the first time that a simple distributed scheduling policy
is proposed that in addition to throughput/utility-optimality exhibits delay
order-optimality.Comment: 44 page
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