Boundary feedback control of Burgers' equation

Consider Burgers' equation

\begin{displaymath} w_{t}-\varepsilon w_{xx}+ww_{x}=0,\qquad x\in \left[ 0,1\right] ,\quad t>0, \end{displaymath} (1)

where \( \varepsilon >0 \) is a constant, with some initial data

\begin{displaymath} w\left( x,0\right) =w_{0}\left( x\right) \, .\end{displaymath}

We achieve set point regulation:

\begin{displaymath} \lim _{t\rightarrow \infty }w\left( x,t\right) =w_{d}\, ,\qquad \forall x\in \left[ 0,1\right] \, ,\end{displaymath}

where \( w_{d} \) is a constant, while keeping \( w\left( x,t\right) \) bounded for all \( \left( x,t\right) \in \left[ 0,1\right] \times \left[ 0,\infty \right) \), using nonlinear Neumann boundary control


$\displaystyle w_{x}\left( 0,t\right)$ $\textstyle =$ $\displaystyle \frac{1}{\varepsilon }\left( c_{0}+\frac{w_{d}}{2}+\frac{1}{9c_{0... ...t\right) -w_{d}\right) ^{2}\right) \left( w\left( 0,t\right) -w_{d}\right) \, ,$ (2)
$\displaystyle w_{x}\left( 1,t\right)$ $\textstyle =$ $\displaystyle -\frac{1}{\varepsilon }\left( c_{1}+\frac{1}{9c_{1}}\left( w\left( 1,t\right) -w_{d}\right) ^{2}\right) \left( w\left( 1,t\right) -w_{d}\right) .,$ (3)

The choice of \( w_{x} \) at the boundary as the control input is motivated by physical considerations. For example in thermal problems one cannot actuate the temperature \( w \), but only the heat flux \( w_{x} \). This makes the stabilization problem nontrivial because homogeneous Neumann boundary conditions make any constant profile an equilibrium solution, thus preventing not only global but even local asymptotic stability. Even mixed linear boundary conditions can introduce multiple stationary solutions.

Simulation Example

We consider first Burgers' equation with zero Neumann boundary condition (uncontrolled system) with \( \varepsilon =0.1 \) and with initial data \( w_{0}=\cos \left( x\right) \) and we regulate first to \( w_{d}=0\) (Figure 1) and then to \( w_{d}=2\) (Figure 2). The uncontrolled system is shown in Figure 1(a) and 2(a). The solution exhibits slow convergence to the zero equilibrium profile. (in fact, for some initial data, the numerical solution gets trapped into this profile and never converges to zero. This unsatisfactory behavior is remedied by applying boundary feedback, as shown in Figure 1(b).

Click on the images to see animation!



Figure 1: Simulations of Burgers' equation with \( w_{d}=0\)
(a) Uncontrolled (b) Controlled, \( c_{0}=c_{1}=10 \)

Figure 2: Simulations of Burgers' equation with \( w_{d}=2\)
(a) Uncontrolled (b) Controlled, \( c_{0}=c_{1}=10 \)

Maintained by Andras Balogh