ccdist2 details:

Keywords: flow control, analytic solution

Global classification: linear-quadratic, convex

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



Submitted on 2013-01-04 by Arnd Rösch. Published on 2013-01-04


ccdist2 description:


Introduction

We have a distributed optimal control problem for the Stokes equations with component-wise box constraints for the distributed control. A difficulty in the construction of test examples as the present one is the satisfaction of the divergence-free condition. Moreover, the example is three-dimensional.

This problem and the analytical solution were published in Rösch and Vexler [2006].

Variables & Notation

Unknowns

q L2(Ω)3 control variable u = (v,p) H01(Ω)3 × L 02(Ω)state variable

Given Data

Ω = (0,1)3 computational domain Γ its boundary a = (0.1,0.1,0.1) lower control bound b = (0.25,0.25,0.01) upper control bound vd,1 = (2 16π2)sin2(πx)sin(2πy)sin(2πz) + (4π2 + 2π)cos(2πx)sin(2πy)sin(2πz) first component of the desired state vd,2 = (1 + 8π2)sin2(πy)sin(2πx)sin(2πz) + (2π2 + 2π)sin(2πx)cos(2πy)sin(2πz) second component of the desired state vd,3 = (1 + 8π2)sin2(πz)sin(2πx)sin(2πy) + (2π2 + 2π)sin(2πx)sin(2πy)cos(2πz) third component of the desired state f1 = 16π2 sin2(πx)sin(2πy)sin(2πz) + (4π2 + 2π)cos(2πx)sin(2πy)sin(2πz) proj[a1,b1](2sin2(πx)sin(2πy)sin(2πz))first component of the given force f2 = 8π2 sin2(πy)sin(2πx)sin(2πz) + (2π2 + 2π)sin(2πx)cos(2πy)sin(2πz) proj[a2,b2](sin2(πy)sin(2πx)sin(2πz)) second component of the given force f3 = 8π2 sin2(πz)sin(2πx)sin(2πy) + (2π2 + 2π)sin(2πx)sin(2πy)cos(2πz) proj[a3,b3](sin2(πz)sin(2πx)sin(2πy)) third component of the given force

Problem Description

Minimize1 2v vdL2(Ω)32 + 1 2qL2(Ω)32 s.t. v + p = q + fin Ω v = 0 in Ω v = 0 on Γ and a q(x) bin Ω.

Optimality System

The following system for the state (velocity and pressure) u = (v,p) H01(Ω)3 × L02(Ω), the control q L2(Ω)3 and the adjoint state z = (w,r) H01(Ω)3 × L02(Ω), given in strong form, characterizes the unique minimizer. The projection is a component-wise projection onto the interval [ai,bi] for i = 1,2,3.

v + p = q + f in Ω v = 0 in Ω v = 0 on Γ w r = v vd in Ω w = 0 in Ω w = 0 on Γ q = proj[a,b](w)in Ω

Supplementary Material

The optimal state, adjoint state and control are known analytically:

v1 = w1 = 2sin2(πx)sin(2πy)sin(2πz) v2 = w2 = sin2(πy)sin(2πx)sin(2πz) v3 = w3 = sin2(πz)sin(2πx)sin(2πy) p = r = sin(2πx)sin(2πy)sin(2πz) q = proj[a,b](w)

References

   A. Rösch and B. Vexler. Optimal control of the Stokes equations: A priori error analysis for finite element discretization with postprocessing. SIAM Journal on Numerical Analysis, 44(5):1903–1920, 2006. doi: 10.1137/050637364.