scdist4 details:

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


scdist4 description:


Introduction

This example is taken from [Cherednichenko et al.2008, Section 5.1]. It features a state constrained problem in which the Lagrange multiplier is given by a Dirac measure.

Variables & Notation

Unknowns

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

Free Parameters

The solution is parametrized in the Tikhonov parameter α > 0.

Given Data

The given data is chosen in a way which admits an analytic solution. This solution is rotationally symmetric.

Ω = B1(0) = {x 2 : |x|2 < 1} computational domain, Γ its boundary, yd(x) = 1 8πα(|x|2 + |x|2 ln|x| + 1) desired state, yc(x) = 1 8πα(2|x| + 1) lower bound.

Problem Description

Minimize 1 2y ydL2(Ω)2 + α 2 uL2(Ω)2 s.t. y = uin Ω, y = 0on Γ, and yc y(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 Lagrange multiplier μ (Ω) = C(Ω¯), given in the strong form, characterizes the unique minimizer.

y = u in Ω, y = 0 on Γ, p = y yd μ in Ω, p = 0 on Γ, αu = p in Ω, μ 0 in (Ω), Ω(y yc)dμ = 0, yc y in Ω¯.

Supplementary Material

The optimal solution together with adjoint state and Lagrange multiplier for the inequality constraint are known. They are given by

y = yd, p = 1 2πln|x|, u = 1 2παln|x|, Ωϕdμ = ϕ(0)ϕ C(Ω¯).

References

   S. Cherednichenko, K. Krumbiegel, and A. Rösch. Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Problems, 24 (5):055003, 21, 2008. doi: 10.1088/0266-5611/24/5/055003.