gcdist1 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 2013-01-14 by Winnifried Wollner. Published on 2013-01-14


gcdist1 description:


Introduction

This is a variation of the mother problem with additional pointwise constraints on the gradient of the state with known analytic solution. The presented problem is given on a domain Ω 2. This problem and analytical solution where proposed in [Deckelnick et al.2008, Section 5], and have been verified in Wollner [2010]. The solution of the problem is special due to the fact that no additional bounds on the control are needed.

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, that is given by rotation of a one dimensional problem.

Ω = B2(0) = {x Ω : |x| 2} computational domain Γ its boundary yΩ(x) = 1 4 + 1 2 ln2 1 4|x|2,0 |x| 1, 1 2 ln2 1 2 ln|x|, 1 < |x| 2. desired state eΩ(x) = 2,0 |x| 1, 0, 1 < |x| 2. given right hand side

Problem Description

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

Optimality System

The following optimality system for the state y H01(Ω) W2,p(Ω) with p > 2, the control u L2(Ω), the adjoint state p Lp (Ω) where 1 p + 1 p = 1, and a Lagrange multiplier μ M(Ω)2 = C(Ω¯)2 for the constraint on the gradient of y characterizes the unique minimizer, see Casas and Fernández [1993]:

y = u + eΩ in Ω y = 0 on Γ p = y yΩ + μin Ω p = 0 on Γ u = p in Ω ϕ y,μC,C 0 ϕ C(Ω¯)2,|ϕ| 1 2, |y| 1 2,μC,C = 0.

Here the adjoint equation has to be understood in the very weak sense, i.e., p solves

Ωpϕdx =Ω(y yΩ)ϕdx +Ωϕdμϕ H01(Ω) C1(Ω¯).

Supplementary Material

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

y = yΩ, p = u, u = 1,0 |x| 1, 0, 1 < |x| 2, μ = y |y|μ0, ϕ,μ0C,C =|x|=1ϕds.

References

   E. Casas and L. A. Fernández. Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Applied Mathematics and Optimization, 27:35–56, 1993. doi: 10.1007/BF01182597.

   K. Deckelnick, A. Günther, and M. Hinze. Finite element approximation of elliptic control problems with constraints on the gradient. Numerische Mathematik, 111: 335–350, 2008. doi: 10.1007/s00211-008-0185-3.

   W. Wollner. A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints. Computational Optimization and Applications, 47(1):133–159, 2010. doi: 10.1007/s10589-008-9209-2.