joint work with Stephanie Thomas
Rate independent systems can be formulated based on an energy functional and a dissipation potential that is assumed to be convex, lower semicontinuous and positively homogeneous of degree one. Here, we will focus on the nonconvex case meaning that the energy is not convex. In this case, the solution typically is discontinuous in time. There exist several (in general not equivalent) notions of weak solutions. We focus on so-called balanced viscosity solutions, discuss the properties of solution sets and discuss the well posedness of an optimal control problem for such systems.
joint work with Christian Clason, Arnd Rösch
This talk is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a $PC^1$ function with countably many points of nondifferentiability, we show that both conditions are equivalent and derive a relaxed optimality system that is amenable to numerical solution using SSN.
This talk is concerned with an optimal control problem governed by a non-smooth coupled system of equations. The non-smooth nonlinearity is Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. We derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative.