scdist5 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 2014-04-04 by Simeon Steinig. Published on 2014-04-05


scdist5 description:


Introduction

Here, we have a distributed optimal control problem of the Poisson equation with pointwise box constraints on the control and a one-sided pointwise state constraint. The present problem is given on the unit ball in Ω = B1(0) 3. The control acts in a distributed way on the entire domain Ω and the state constraint is enforced on the entire domain, too. This problem and the analytical solution appear in [Rösch and Steinig2012, Section 8]. The problem is designed to have a vanishing Lagrange multiplier for the state constraint. It is thus potentially a good test case for a posteriori error estimation.

Variables & Notation

Unknowns

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

Given Data

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

Ω = B1(0) 3 computational domain Ω its boundary yd(x) = 4π2|x|2 sin(π|x|2) + 6πcos(π|x|2) + cos(π 2|x|2) desired state f(x) = 3πsin(π 2|x|2) + π2|x|2 cos(π 2|x|2) + sin(π|x|2) source shift yc(x) = cos(π 2 |x|2) if |x| 0.5 (4 3 cos(π 8 ) 40 3 )|x|2 + 4 3 cos(π 8 ) + 10 3 else state constraint

Problem Description

Minimize 1 2y ydL2(Ω)2 + 1 2uL2(Ω)2 s.t. y = u + fin Ω y = 0 on Ω and 1 u(x) 1in Ω. and yc(x) y(x)in Ω

Optimality System

The following optimality system, given in the strong form, for the state y H01(Ω), the control u L2(Ω), the adjoint state p H01(Ω) and the Lagrange multiplier μ (Ω) characterizes the unique minimizer.

y = u + f in Ω, y = 0 on Ω, p = y yd μ in Ω, p = 0 on Ω, u = proj[1,1](p) in Ω, μ,y ycC(Ω ̄),C(Ω ̄) = 0, μ 0, yc y.

Supplementary Material

The optimal state and control are known analytically:

y = cos(π 2|x|2), p = sin(π|x|2), u = sin(π|x|2), μ = 0.

Note that the state constraint is active on the ball |x| 1 2. This means that strict complementarity fails on the active set, a fact which makes the problem numerically challenging.

References

   A. Rösch and S Steinig. A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM: Mathematical Modelling and Numerical Analysis, 46(5):1107–1120, 2012. doi: 10.1051/m2an/2011076.