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Keywords: analytic solution
Global classification: nonlinear-quadratic
Functional: convex quadratic
Geometry: easy, fixed
Design: coupled via boundary values 1st order
- Quasi-linear elliptic:
- quasi-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 2013-02-05 by Roland Herzog. Published on 2013-02-12
This is a boundary optimal control problem governed by a quasilinear elliptic equation. Problems involving quasilinear equations are particularly challenging with regard to their analysis, numerical analysis, and numerical solution.
The given data is chosen in a way which admits an analytic solution.
Note that the formula for the auxiliary term above has been corrected (a square was missing). The formula originally given in [Casas and Dhamo, 2012, p.753] and [Dhamo, 2012, p.117] does not match the function shown in [Dhamo, 2012, Figure 4.1]. Preference has been given to the reproduction of the ﬁgure.
The following optimality system for the state , the control and the adjoint state , given in the strong form, represents a set of ﬁrst-order necessary optimality conditions.
The following state, control and adjoint state variables are shown in Casas and Dhamo  to satisfy ﬁrst-order necessary conditions. Moreover, second-order suﬃcient conditions also hold due to the structure of the objective. Consequently, is a local minimum (in the sense of ).
This solution is particular in the sense that the upper and lower bound constraints for the control are both active on nontrivial parts of the boundary, but never strongly active. In other words, strict comlementarity does not hold for this problem. A plot of the optimal control is provided in [Dhamo, 2012, Figure 4.1].
E. Casas and V. Dhamo. Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations. Computational Optimization and Applications. An International Journal, 52(3): 719–756, 2012. ISSN 0926-6003. doi: 10.1007/s10589-011-9440-0.
V. Dhamo. Optimal Boundary Control of Quasilinear Partial Diﬀerential Equations: Theory and Numerical Analysis. PhD thesis, Technische Universität Berlin, 2012. URL http://opus.kobv.de/tuberlin/volltexte/2012/3511.