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 3-dim domain in 3-dim space
- No time dependence.
- box of order 0
- box of order 0
Submitted on 2014-04-04 by Simeon Steinig. Published on 2014-04-05
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 . The control acts in a distributed way on the entire domain and the state constraint is enforced on the entire domain, too. This problem and the analytical solution appear in [Rösch and Steinig, 2012, 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.
The given data is chosen in a way which admits an analytic solution.
The following optimality system, given in the strong form, for the state , the control , the adjoint state and the Lagrange multiplier characterizes the unique minimizer.
The optimal state and control are known analytically:
Note that the state constraint is active on the ball . This means that strict complementarity fails on the active set, a fact which makes the problem numerically challenging.
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.