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:


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

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 < 00 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.

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.