A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

Samuel Amstutz (1) , Antoine Laurain (2)

1 LANLG - Laboratoire d'Analyse non linéaire et Géométrie
2 Institut für Mathematik [Berlin]

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints $0 \leq u \leq 1$ and $\int u=m$. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take $0-1$ values, which amounts to solving a topology optimization problem with volume constraint.

Mots clés: optimal control; topology optimization; semilinear equation; semismooth Newton method; volume constraint

Domaines: Optimisation et contrôle [math.OC]

Samuel Amstutz, Antoine Laurain. A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints. 2011. ⟨hal-00636063v2⟩