Distributed Control of the Generalized Korteweg-de Vries-Burgers Equation

The paper deals with the distributed control of the generalized Kortweg-de Vries-Burgers equation (GKdVB) subject to periodic boundary conditions via the Karhunen-Loève (K-L) Galerkin method. The decomposition procedure of the K-L method is presented to illustrate the use of this method in analyzing the numerical simulations data which represent the solutions to the GKdVB equation. The K-L Galerkin projection is used as a model reduction technique for nonlinear systems to derive a system of ordinary differential equations (ODEs) that mimics the dynamics of the GKdVB equation. The data coefficients derived from the ODE system are then used to approximate the solutions of the GKdVB equation. Finally, three state feedback linearization control schemes with the objective of enhancing the stability of the GKdVB equation are proposed. Simulations of the controlled system are given to illustrate the developed theory

Controlling the dynamics of Burgers equation with a high-order nonlinearity

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e., ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball in L2[0,1] and uniqueness of steady state solutions for all integer n≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results

A hybrid neural network model for the dynamics of the Kuramoto-Sivashinsky equation

A hybrid approach consisting of two neural networks is used to model the oscillatory dynamical behavior of the Kuramoto-Sivashinsky (KS) equation at a bifurcation parameter α=84.25. This oscillatory behavior results from a fixed point that occurs at α=72 having a shape of two-humped curve that becomes unstable and undergoes a Hopf bifurcation at α=83.75. First, Karhunen-Loève (KL) decomposition was used to extract five coherent structures of the oscillatory behavior capturing almost 100% of the energy. Based on the five coherent structures, a system offive ordinary differential equations (ODEs) whose dynamics is similar to the original dynamics of the KS equation was derived via KL Galerkin projection. Then, an autoassociative neural network was utilized on the amplitudes of the ODEs system with the task of reducing the dimension of the dynamical behavior to its intrinsic dimension, and a feedforward neural network was usedto model the dynamics at a future time. We show that by combining KL decomposition and neural networks, a reduced dynamical model of the KS equation is obtained

Symmetries, Dynamics, and Control for the 2D Kolmogorov Flow

The symmetries, dynamics, and control problem of the two-dimensional (2D) Kolmogorov flow are addressed. The 2D Kolmogorov flow is known as the 2D Navier-Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the x-direction. First, using the Fourier Galerkin method on the original 2D Navier-Stokes equations, we obtain a seventh-order system of nonlinear ordinary differential equations (ODEs) which approximates the behavior of the Kolmogorov flow. The dynamics and symmetries of the reduced seventh-order ODE system are analyzed through computer simulations for the Reynolds number range 0<Re<26.41. Extensive numerical simulations show that the obtained system is able to display the different behaviors of the Kolmogorov flow. Then, we design Lyapunov based controllers to control the dynamics of the system of ODEs to different attractors (e.g., a fixed point, a periodic orbit, or a chaotic attractor). Finally, numerical simulations are undertaken to validate the theoretical developments

The generalized Burgers equation with and without a time delay

We consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2π. For the generalized Burgers equation without a time delay, that is, ut=vuxx−uux+u+h(x), 0<x<2π, t>0, u(0,t)=u(2π,t), u(x,0)=u0(x), a Lyapunov function method is used to show boundedness and uniqueness of a steady state solution and global stability of the equation. As for the generalized time-delayed Burgers equation, that is, ut(x,t)=vuxx(x,t)−u(x,t−τ)ux(x,t)+u(x,t), 0<x<2π, t>0, u(0,t)=u(2π,t), t>0, u(x,s)=u0(x,s), 0<x<2π, −τ≤s≤0, we show that the equation is exponentially stable under small delays. Using a pseudospectral method, we present some numerical results illustrating and reinforcing the analytical results

Time-Delayed Feedback Control of a Hydraulic Model Governed by a Diffusive Wave System

This paper is concerned with the feedback flow control of an open-channel hydraulic system modeled by a diffusive wave equation with delay. Firstly, we put forward a feedback flow control subject to the action of a constant time delay. Thereafter, we invoke semigroup theory to substantiate that the closed-loop system has a unique solution in an energy space. Subsequently, we deal with the eigenvalue problem of the system. More importantly, exponential decay of solutions of the closed-loop system is derived provided that the feedback gain of the control is bounded. Finally, the theoretical findings are validated via a set of numerical results