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Mathematical control of complex systems
Copyright © 2013 ZidongWang et al.This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers
In this paper we introduce a finite-parameters feedback control algorithm for
stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped
nonlinear wave equations and the nonlinear wave equation with nonlinear damping
term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation.
This algorithm capitalizes on the fact that such infinite-dimensional
dissipative dynamical systems posses finite-dimensional long-time behavior
which is represented by, for instance, the finitely many determining parameters
of their long-time dynamics, such as determining Fourier modes, determining
volume elements, determining nodes , etc..The algorithm utilizes these finite
parameters in the form of feedback control to stabilize the relevant solutions.
For the sake of clarity, and in order to fix ideas, we focus in this work on
the case of low Fourier modes feedback controller, however, our results and
tools are equally valid for using other feedback controllers employing other
spatial coarse mesh interpolants
Stabilization of a linear Korteweg-de Vries equation with a saturated internal control
This article deals with the design of saturated controls in the context of
partial differential equations. It is focused on a linear Korteweg-de Vries
equation, which is a mathematical model of waves on shallow water surfaces. In
this article, we close the loop with a saturating input that renders the
equation nonlinear. The well-posedness is proven thanks to the nonlinear
semigroup theory. The proof of the asymptotic stability of the closed-loop
system uses a Lyapunov function.Comment: European Control Conference, Jul 2015, Linz, Austri
Global sensitivity analysis for the boundary control of an open channel
The goal of this paper is to solve the global sensitivity analysis for a
particular control problem. More precisely, the boundary control problem of an
open-water channel is considered, where the boundary conditions are defined by
the position of a down stream overflow gate and an upper stream underflow gate.
The dynamics of the water depth and of the water velocity are described by the
Shallow Water equations, taking into account the bottom and friction slopes.
Since some physical parameters are unknown, a stabilizing boundary control is
first computed for their nominal values, and then a sensitivity anal-ysis is
performed to measure the impact of the uncertainty in the parameters on a given
to-be-controlled output. The unknown physical parameters are de-scribed by some
probability distribution functions. Numerical simulations are performed to
measure the first-order and total sensitivity indices
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