ccdist1 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 2012-08-21 by Roland Herzog. Published on 2012-08-21


ccdist1 description:


Introduction

Here we have a simple distributed optimal control problem of the Poisson equation with a potential term, and with pointwise bound constraints on the control. Problems of this type are treated extensively in [Tröltzsch2010, Chapter 2], and are sometimes refered to as the mother problem type. The present problem is special in the sense that the control acts in a distributed way on the entire domain Ω, and that the state is observed on the entire domain as well. Moreover, the non-trivial part of the solution is rotationally symmetric.

This problem and analytical solution appear in [Tröltzsch2010, Section 2.9.2].

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. This solution is rotationally symmetric on a circle of radius 1/3 strictly contained in Ω.

Ω = (0,1)2 computational domain Γ its boundary x̂ = (0.5,0.5) center of Ω yΩ = 142 3 + 12x x̂2 desired state eΩ = 1 proj[0,1]12x x̂2 1 3uncontrolled force eΓ = 12 boundary observation coefficient

Problem Description

Minimize1 2y yΩL2(Ω)2 +ΓeΓyds + 1 2uL2(Ω)2 s.t. y + y = u + eΩin Ω y n = 0 on Γ and 0 u(x) 1in Ω.

Optimality System

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

y + y = u + f in Ω y n = 0 on Γ p + p = y yΩ in Ω p n = eΓ on Γ u = proj[0,1](p)in Ω

Supplementary Material

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

y = 1 p = 12x x̂2 + 1 3 u = 1, x Ω3 12x x̂2 13,x Ω2 0, x Ω1

where the subdomains are defined as follows:

Ω1 = {x Ω : x x̂ < 16} Ω2 = {x Ω : 16 x x̂ < 13} Ω3 = {x Ω : 13 x x̂}.

References

   F. Tröltzsch. Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, 2010.