Welcome to the OPTPDE Problem Collection
Keywords: analytic solution
Global classification: linear-quadratic, convex
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via volume data
- linear elliptic operator of order 2.
- Defined on a 2-dim domain in 2-dim space
- No time dependence.
- box of order 0
Submitted on 2012-08-21 by Roland Herzog. Published on 2012-08-21
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 [Tröltzsch, 2010, 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 , and that the state is observed on the entire domain as well. Moreover, the non-trivial part of the solution is rotationally symmetric.
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 .
The following optimality system for the state , the control and the adjoint state , given in the strong form, characterizes the unique minimizer.
The optimal state, adjoint state and control are known analytically:
where the subdomains are deﬁned as follows: