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hypini1 details:

Keywords: flow control, analytic solution

Global classification: nonlinear-quadratic

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via initial data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:

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

hypini1 description:


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


u = (u0,u,ur) L(Ω) × × control variable y L( × (0,T)) state variable

Given Data

Ω = (4,4) computational domain Ω = (,4] outer left domain Ω = [4,) outer right domain T = 1 final time yΩ = 1if x < 0 0 if  x 0 desired state

Problem Description

Minimizey(,T) yΩL2(Ω)2 s.t. y is a weak entropy solution of ty + x y2 2 = 0 in  × (0,T) y(x,0) = u if x Ω u0(x)if x Ω ur if x Ωr on .

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 × (0,T), the equation can be restricted to Ω × (0,T), and then u and ur can be described as constant boundary data at x {4,4}. This leads to a very similar problem with the same optimal solution.

Supplementary Material

A globally optimal state and control are known analytically:

y = 1,x < t1 2 , 0,x t1 2 , u0 = 1,x < 1 2, 0,x 1 2, u = 1, ur = 0.

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 Ω within the time interval (0,1).

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

u0 = 2,x < 1 4, 0,x 1 4, , u = 2, ur = 0 as an initial guess for the control variables as their Experiment 1 and the alternative initial guess u0 = 1, u = 1, ur = 1.

as their Experiment 2.


   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.