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Keywords: analytic solution
Global classification: linear-quadratic, convex
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via boundary values 1st order
- 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 2013-01-09 by Arnd Rösch. Published on 2013-01-13
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 of the circular sector.
The given data is chosen in a way which admits an analytic solution. The domain and the solution depend on the angle . The most interesting cases arise when is the largest angle, i.e., in case . The optimal control has the natural low regularity described by the singular exponent , which also depends on .
The description of the problem is most convenient when both cartesian coordinates and polar coordinates are used interchangeably.
The function can be computed by the formula
with denoting the outer normal vector to , and
Note that vanishes at the part of that coincides with the boundary of the circular sector.
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:
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.