ccbnd1 details:

Keywords: analytic solution

Global classification: linear-quadratic, convex

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via boundary values 1st order

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



Submitted on 2013-01-09 by Arnd Rösch. Published on 2013-01-13


ccbnd1 description:


Introduction

Here we have a simple boundary optimal control problem of the Poisson equation with pointwise box constraints on the control. The domain is polygonal, and it is the intersection of a square and a circular sector. The regularity of the optimal solution and consequently the approximation properties of numerical solutions depend on the angle ω of the circular sector.

This problem and the analytical example were published in Mateos and Rösch [2011].

Variables & Notation

Unknowns

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

Given Data

The given data is chosen in a way which admits an analytic solution. The domain Ωω and the solution depend on the angle 0 < ω < 2π. The most interesting cases arise when ω is the largest angle, i.e., in case ω π2. The optimal control has the natural low regularity described by the singular exponent λ, which also depends on ω.

The description of the problem is most convenient when both cartesian coordinates (x1,x2) and polar coordinates (r,ϕ) are used interchangeably.

Sω = {(rcosϕ,rsinϕ) : r [0,2),ϕ (0,ω)}circular sector Ωω = (1,1)2 S ω computational domain Γω its boundary λ = πω singular exponent yd(r,ϕ) = rλ cos(λϕ) desired state (polar coordinates) g1(x1,x2) = nyd(x1,x2) objective term g2(x1,x2) = proj[0.5,0.5]yd(x1,x2) boundary term

The function g1 can be computed by the formula

g1 = nyd = yd n

with n denoting the outer normal vector to Γω, and

yd = yd x1 yd x2 = λrλ1 cos(λϕ)x1 r λrλ sin(λϕ)x2 r2 λrλ1 cos(λϕ)x2 r + λrλ sin(λϕ)x1 r2 .

Note that g1 vanishes at the part of Γω that coincides with the boundary of the circular sector.

Problem Description

Minimize1 2y ydL2(Ωω)2 +Γωg1yds + 1 2uL2(Γω)2 s.t. y + y = 0 in Ωω y n = u + g2on Γω and 0.5 u(x1,x2) 0.5on Γω.

Optimality System

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

y + y = 0 in Ωω y n = u + g2 on Γω p + p = y yd in Ωω p n = g1 on Γω u = proj[0.5,0.5](p|Γω)on Γω

Supplementary Material

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

y = 0 in Ωω p = ydin Ωω u = g2on Γω

References

   M. Mateos and A. Rösch. On saturation effects in the Neumann boundary control of elliptic optimal control problems. Computational Optimization and Applications, 49 (2):359–378, 2011. doi: 10.1007/s10589-009-9299-5.