Boundary control of the Korteweg-de Vries-Burgers equation: further results on stabilization and well-posedness, with numerical demonstration

@article{Balogh2000BoundaryCO,
  title={Boundary control of the Korteweg-de Vries-Burgers equation: further results on stabilization and well-posedness, with numerical demonstration},
  author={Andr{\'a}s Balogh and Miroslav Krsti{\'c}},
  journal={IEEE Trans. Autom. Control.},
  year={2000},
  volume={45},
  pages={1739-1745}
}
We consider the Korteweg-de Vries-Burgers equation on the interval [0,1]. Motivated by simulations resulting in modest decay rates with recently proposed control laws by Liu and Krstic, which keeps some of the boundary conditions as homogeneous, we propose a strengthened set of feedback boundary conditions. We establish stability properties of the closed-loop system, prove well-posedness and illustrate the performance improvement by a simulation example. 

Figures from this paper

Stability of Burgers-Korteweg-de Vries Equation
  • Wenhua Gao, F. Deng
  • Mathematics
    2007 IEEE International Conference on Control and Automation
  • 2007
For Burgers-Korteweg-de Vries(KdVB) equation defined on a finite spatial interval with unknown viscosity, a nonlinear boundary control law and an adaptation law are proposed. Based on Lyapunov direct
Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation
In this paper, we consider the boundary control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three control laws are derived for
Robust stabilization the Korteweg–de Vries–Burgers equation by boundary control
The problem of robust global stabilization by nonlinear boundary feedback control for the Korteweg–de Vries–Burgers equation on the domain [0,1] is considered. The main purpose of this paper is to
Stabilization of the Korteweg-de Vries-Burgers equation with non-periodic boundary feedbacks
We consider the Korteweg-de Vries-Burgers (KdVB) equation on the domain [0,1]. We derive a control law which guaranteesL2-global exponential stability,H3-global asymptotic stability, andH3-semiglobal
Boosting the decay of solutions of the linearised Korteweg-de Vries–Burgers equation to a predetermined rate from the boundary
TLDR
It is proven that all the sufficiently small solutions can be steered to zero at any desired exponential rate by means of a suitably constructed boundary feedback controller.
Modelling the dynamics of the Generalized Korteweg-de Vries-Burgers equation using boundary control
The nonlinear boundary control problem of the Generalized Korteweg-de Vrie-Burgers (GKdVB) equation: ut = vuxx -- μuxxx -- uαux, x ∈ [0, 1], t > 0 is considered. Different control laws were derived
Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg-de Vries-Burgers Equation
TLDR
Using Lyapunov theory, the - global exponential stability of the solution is proven in each case of the modified generalized Korteweg–de Vries–Burgers equation when the spatial domain is .
Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation
In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is [0,1]$[0,1]$.
...
...

References

SHOWING 1-10 OF 23 REFERENCES
Global Boundary Stabilization of the Korteweg-de Vries-Burgers Equation
The problem of global exponential stabilization by boundary feedback for the Korteweg-de Vries-Burgers equation on the domain [0, 1] is considered. We derive a control law of the form u(0) = ux(1) =
Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain
The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial
The initial-value problem for the Korteweg-de Vries equation
  • J. Bona, R. Smith
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1975
For the Korteweg-de Vries equation ut+ux+uux+uxxx=0, existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞<x<∞) and
On the Global Dynamics of a Controlled Viscous Burgers' Equation
In this paper we consider a boundary control problem for a forced Burgers' equation in a Hilbert state space consisting of square integrable functions on a finite interval. Our first main result
Stabilization of the Korteweg-de Vries Equation on a Periodic Domain
We study solutions of the Korteweg-de Vries (KdV) equations $$ {u_t} + u{u_x} + {u_{xxx}} = f $$ and $$ {u_t} + u{u_x} + {u_{xxx}} = 0 $$ for t ≥ 0 and 0 ≤ x ≤ 1 where the subscripts
Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping
A semilinear model of the wave equation with nonlinear boundary conditions and nonlinear boundary velocity feedback is considered. Under the assumption that the velocity boundary feedback is
Computational methods for fluid dynamics
This text develops and applies the techniques used to solve problems in fluid mechanics on computers and describes in detail those most often used in practice. It includes advanced techniques in
...
...