Departmentof Mathematics

Introduction

This is an optimal control problem for entropy solutions of the inviscid one-dimensional Burgers equation considered in Castro et al. [2008]. The control acts as initial data and the objective function is a tracking type functional at end time with discontinuous desired state.

Variables & Notation

Unknowns

$$\begin{array}{ccc}\hfill u=\left(u_0,u_{\ell },u_r\right)\in L^{\infty }\left(\Omega \right)\times ℝ\times ℝ& \text{control variable}\hfill & \hfill \\ \hfill y\in L^{\infty }\left(ℝ\times \left(0,T\right)\right)& \text{state variable}\hfill \end{array}$$

Given Data

$$\begin{array}{ccccc}\hfill \Omega & =\left(-4,4\right)\hfill & \hfill & \text{computational domain}\hfill & \hfill \\ \hfill {\Omega }_{\ell }& =\left(-\infty ,-4\right]\hfill & \hfill & \text{outer left domain}\hfill \\ \hfill {\Omega }_{\ell }& =\left[4,\infty \right)\hfill & \hfill & \text{outer right domain}\hfill \\ \hfill T& =1\hfill & \hfill & \text{final time}\hfill \\ \hfill y_{\Omega }& =\left\{\begin{array}{cc}1\hfill & \text{if }x<0\hfill \\ 0\hfill & \text{if }x\ge 0\hfill \end{array}\right.\hfill & \hfill & \text{desired state}\hfill \\ \hfill \end{array}$$

Problem Description

$$\begin{array}{ccc}\hfill \text{Minimize}& \parallel y\left(\cdot ,T\right)-y_{\Omega }{\parallel }_{L^2\left(\Omega \right)}^2\hfill & \hfill \\ \hfill \text{s.t.}& y\text{ is a weak entropy solution of}\hfill \\ \hfill & \left\{\begin{array}{ccccc}\hfill & \frac{\partial }{\partial t}y+\frac{\partial }{\partial x}\left(\frac{y^2}{2}\right)=0\hfill & \hfill & \text{in }ℝ\times \left(0,T\right)\hfill & \hfill \\ \hfill & y\left(x,0\right)=\left\{\begin{array}{cc}{u}_{\ell }\hfill & \text{if }x\in {\Omega }_{\ell }\hfill \\ u_0\left(x\right)\hfill & \text{if }x\in \Omega \hfill \\ u_r\hfill & \text{if }x\in {\Omega }_r\hfill \end{array}\right.\hfill & \hfill & \text{on }ℝ.\hfill \end{array}\right.\hfill \end{array}$$

This problem appears as [Castro et al., 2008, Section 7, Experiment 1]. Suitable numerical schemes for the Burgers equation are given in [Castro et al., 2008, Section 3]. Note that the state is expected to have shock discontinuities.

Instead of formulating the Burgers equation on $ℝ\times \left(0,T\right)$, the equation can be restricted to $\Omega \times \left(0,T\right)$, and then $u_{\ell }$ and $u_r$ can be described as constant boundary data at $x\in \left\{-4,4\right\}$. This leads to a very similar problem with the same optimal solution.

Supplementary Material

A globally optimal state and control are known analytically:

$$\begin{array}{ccc}\hfill y& =\left\{\begin{array}{cc}1,\hfill & x<\frac{t-1}{2},\hfill \\ 0,\hfill & x\ge \frac{t-1}{2},\hfill \end{array}\right.\hfill & \hfill \\ \hfill u_0& =\left\{\begin{array}{cc}1,\hfill & x<-\frac{1}{2},\hfill \\ 0,\hfill & x\ge -\frac{1}{2},\hfill \end{array}\right.\hfill \\ \hfill u_{\ell }& =1,\hfill \\ \hfill u_r& =0.\hfill \end{array}$$

Note that the value of the objective is zero for this solution. The optimal state contains a shock which moves through the domain but does not reach the boundary of $\Omega$ within the time interval $\left(0,1\right)$.

The performance of a gradient descent method and an alternating descent method in combination with various discretizations of the state equation is described in [Castro et al., 2008, Section 7]. The authors use

$$\begin{array}{llllllllllll}\hfill u_0& =\left\{\begin{array}{cc}2,\hfill & x<\frac{1}{4},\hfill \\ 0,\hfill & x\ge \frac{1}{4},\hfill \end{array}\right.,& \hfill u_{\ell }& =2,& \hfill u_r& =0& \hfill & & \hfill & & \hfill & \\ \multicolumn{12}{c}{\text{as an initial guess for the control variables as their Experiment 1 and the alternative initial guess}}\\ \\ \hfill u_0& =-1,& \hfill u_{\ell }& =-1,& \hfill u_r& =-1.& \hfill & & \hfill & & \hfill & \end{array}$$

as their Experiment 2.

References