## hypini1 details:

Keywords: flow control, analytic solution

Geometry: easy, fixed

Design: coupled via initial data

Differential operator:

• Burger:
• nonlinear hyperbolic operator of order 1.
• Defined on a 1-dim domain in 1-dim space
• Time dependent.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2013-02-20 by Stefan Ulbrich. Published on 2013-02-24

## hypini1 description:

### 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.

### Problem Description

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 $ℝ×\left(0,T\right)$, the equation can be restricted to $\Omega ×\left(0,T\right)$, 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:

 $\begin{array}{ccc}\hfill y& =\left\{\begin{array}{cc}1,\phantom{\rule{1em}{0ex}}\hfill & x<\frac{t-1}{2},\hfill \\ 0,\phantom{\rule{1em}{0ex}}\hfill & x\ge \frac{t-1}{2},\hfill \end{array}\right\\hfill & \hfill \\ \hfill {u}_{0}& =\left\{\begin{array}{cc}1,\phantom{\rule{1em}{0ex}}\hfill & x<-\frac{1}{2},\hfill \\ 0,\phantom{\rule{1em}{0ex}}\hfill & x\ge -\frac{1}{2},\hfill \end{array}\right\\hfill \\ \hfill {u}_{\ell }& =1,\hfill \\ \hfill {u}_{r}& =0.\hfill \end{array}$

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

as their Experiment 2.

### References

C. Castro, F. Palacios, and E. Zuazua. An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks. Mathematical Models & Methods in Applied Sciences, 18(3):369–416, 2008. doi: 10.1142/S0218202508002723.