scdist2 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


scdist2 description:


Introduction

This problem was introduced in [Benedix and Vexler2009, Example 1]. The example is constructed such that the analytic solution in known, and the Lagrange multiplier for the pointwise state constraints is given by a line measure and a volume contribution using smooth problem data. The example is parametrized by three real numbers that allow to steer the contributions of the volume and line measure. Further, the position of the line measure can be chosen to avoid that it coincides with edges of the discretization.

Variables & Notation

Unknowns

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

Free parameters

Within the problem, there are three free parameters (s,m,b) that can be chosen by the user:

The calculations in Benedix and Vexler [2009] were conducted using b = 50, m = 2, and s = 0.125 as well as s = 0.3.

Given Data

The given data is chosen in a way which admits an analytic known solution. The solution is constant along the x2 direction.

Ω = (0,1)2 computational domain, Γ its boundary, Γ1 = {x = (x1,x2) Γ|x1 = 0} Dirichlet boundary, Γ2 = Γ Γ1 Neumann boundary.

With this one defines

HD1(Ω) = {v H1(Ω)|v = 0 on Γ 1}.

The desired state is given by

ud(x1,x2) = x13s3 3x12s2 + 3x1s1 + 2, x1 < s, 3m 4(1s)(x1 s)4 + m(x1 s)3 + 3,x1 s.

The upper bound is given by

ub(x1,x2) = 1, x1 < s, 3m 4(1s)(x1 s)4 + m(x1 s)3 + 1,x1 s.

Moreover, the uncontrolled force is set to be

f(x1,x2) = 6s2 6mx1 + x1(x1 2) + b(1 s)x1, x1 < s, (1 r)x12 + (b 18ms 1s 2 6m)x1 + 6s2 rs2,x1 s,

where

r = b 2 9m 1 s.

Problem Description

Minimize 1 2u udL2(Ω)2 + 1 2qL2(Ω)2 s.t. u = q + fin Ω, u = 0 on Γ1, u n = 0 on Γ2, and u(x) ubin Ω¯.

Optimality System

The following optimality system for the state u HD1(Ω), the control q L2(Ω), the adjoint state z HD1(Ω), and the Lagrange multiplier for the inequality constraints μ (Ω) = C(Ω¯) given in the strong form, characterizes the unique minimizer.

u = q + f in Ω, u = 0 on Γ1, u n = 0 on Γ2, z = u ud + μ in Ω, z = 0 on Γ1, z n = 0 on Γ2, q = z in Ω, μ 0 in (Ω), Ω(u ub)dμ = 0, u ub on Ω¯.

Supplementary Material

The optimal solution is known analytically. It is given by

u(x1,x2) = x13s3 3x12s2 + 3x1s1,x1 < s, ub(x1,x2), x1 s, q(x1,x2) = x1(x1 2) (6s3 6m)x1 b(1 s)x1, x1 < s, x1(x1 2) 6s2 + 6ms + b 2x12 bx1 + b 2s2,x1 s, z(x1,x2) = q(x1,x2) μ = μ1 + μ2 Ωϕdμ1 =( 6 s3 6m)01ϕ(s,x 2)dx2ϕ C(Ω¯), μ2(x1,x2) = 0,x1 < s, b, x1 s.

References

   O. Benedix and B. Vexler. A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Computational Optimization and Applications, 44(1):3–25, 2009. doi: 10.1007/s10589-008-9200-y.