Semi-smooth Newton methods for optimal control of the dynamical Lamé system with control constraints.

Axel Kröner (1, 2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems

Optimal control problems governed by the dynamical Lamé system with additional constraints on the controls are analyzed. Different types of control action are considered: distributed, Neumann boundary and Dirichlet boundary control. To treat the inequality control constraints semi-smooth Newton methods are applied and their convergence is analyzed. Although semi-smooth Newton methods are widely studied in the context of PDE-constrained optimization little has been done in the context of the dynamical Lamé system. The novelty of the article is the proof that in case of distributed and Neumann boundary control the Newton method converges superlinearly. In case of Dirichlet control superlinear convergence is shown for a strongly damped Lamé system. The results are an extension of [A. Kröner , K. Kunisch , and B. Vexler ( 2011 ), SIAM J. Control Optim. 49 : 830 – 858], where optimal control problems of the classical wave equation are considered. The control problems are discretized by finite elements and numerical examples are presented.

Mots clés: control constraints; dynamical Lamé system; finite elements; optimal control; semi-smooth Newton methods

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]

Axel Kröner. Semi-smooth Newton methods for optimal control of the dynamical Lamé system with control constraints.. Numerical Functional Analysis and Optimization, 2013. ⟨hal-01383266⟩