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Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations

Casas E., Dhamo V.

Control and Cybernetics

2011

Vol. 40, no 2

457-490

First- and second-order optimality conditions are established for the boundary optimal control of quasilinear elliptic equations with pointwise constraints on the control. The theory is developed for Neumann controls in polygonal domains of dimension two. For the derivation of second-order sufficient optimality conditions, which is the main goal of this paper, the regularity of the solutions to the state equation and its linearization is studied in detail. Moreover, a Pontryagin principle is proved. The main difficulty in the analysis of these problems is the nonmonotone character of the state equation.

A note on variational discretization of elliptic Neumann boundary control

Hinze M., Matthes U.

Control and Cybernetics

2009

Vol. 38, no 3

577-591

We consider variational discretization of Neumann-type elliptic optimal control problems with constraints on the control. In this approach the cost functional is approximated by a sequence of functionals, which are obtained by discretizing the state equation with the help of linear finite elements. The control variable is not discretized. Error bounds for control and state are obtained both in two and three space dimensions. Finally, we discuss some implementation issues of a generalized Newton method applied to the numerical solution of the problem class under consideration.