A control problem approach to Coulomb’s friction

In this work we present a formulation of Coulomb's friction in a fractured elastic body as a PDE control problem where the observed quantity is the tangential stress across an internal interface, while the control parameter is the slip i.e. the displacement jump across the interface. The cost function aims at minimizing the norm of a non-linear and not everywhere differentiable complementarity function, written in terms of the tangential stress and the slip. The interesting point of this method is that gives rise to an iterative procedure where at each iteration we solve a problem with given slip at the interface, without resorting to the use of Lagrange multipliers. We carry out a formal derivation of the method, with some preliminary results, and a numerical experiment to verify the efficacy of the technique.