scdist3 details:

Keywords:

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 2014-02-15 by Winnifried Wollner. Published on 2017-01-09


scdist3 description:


Introduction

This example is taken from Günther and Hinze [2008]. It features a complex active set structure for the inequality constraints on the state.

Variables & Notation

Unknowns

u L2(Ω) control variable y H1(Ω)state variable

Given Data

Ω = (0,1)2 computational domain Γ its boundary u0 = 60 desired control y0 = 0.5 desired state a = 0.45 lower bound for the state b(x1,x2) = min1,max0.5,50(x1 0.3)2 + (x 2 0.3)2upper bound for the state

Problem Description

Minimize 1 2y y0L2(Ω)2 + 1 2u u0L2(Ω)2 s.t. y + y = uin Ω y n = 0on Γ and a y(x) b(x)in Ω¯.

Optimality System

The following optimality system for the state y H01(Ω), the control u L2(Ω), the adjoint state p H01(Ω), and the Lagrange multipliers μa,μb (Ω) = C(Ω¯) for the lower and upper inequality constraint, respectively, given in the strong form, characterizes the unique minimizer.

y + y = u in Ω, y n = 0 on Γ, p + p = y y0 + μb μa in Ω, p n = 0 on Γ, u = u0 p, μa 0, μb 0, Ω(a y)dμa = 0, Ω(y b)dμb = 0, a y b.

Supplementary Material

A reference value for the functional is provided in Günther and Hinze [2008] as

J 1759.04686

References

   A. Günther and M. Hinze. A-posteriori error control of a state constrained elliptic control problem. Journal of Numerical Mathematics, 16:307–322, 2008. doi: 10.1515/JNUM.2008.014.