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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.
- nonlinear convex, local of order 1
Submitted on 2013-01-14 by Winnifried Wollner. Published on 2013-01-14
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 . This problem and analytical solution where proposed in [Deckelnick et al., 2008, Section 5], and have been veriﬁed in Wollner . The solution of the problem is special due to the fact that no additional bounds on the control are needed.
The given data is chosen in a way which admits an analytic solution, that is given by rotation of a one dimensional problem.
The following optimality system for the state with , the control , the adjoint state where , and a Lagrange multiplier for the constraint on the gradient of characterizes the unique minimizer, see Casas and Fernández :
Here the adjoint equation has to be understood in the very weak sense, i.e., solves
The optimal state, adjoint state, control and Lagrange multiplier are known analytically:
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.