Workpackage 1

The main focus of Workpackage 1 is on optimization and optimal control problems involving the L^p functional  with p ∈ (0,1).
Our first goal is the development of new suitable analytical and numerical concepts to overcome the difficulties caused by the lack of Lipschitz continuity and of convexity, for which we cannot rely on the existing techniques (involving e.g. lower-semi-continuity or reflexivity of Banach spaces). Then we aim at exploiting the use of non-smooth and non-convex functionals first in the context of optimal control, where it is of relevant importance for  sparsity of solutions, switching controls and optimal actuator placement.  While we first study L^p functionals in the context of open loop optimal control (Workpackage 1), they are equally important and novel for closed loop problems and therefore will be considered also in Workpackages 3 and 4.
We will also show how the use of these functionals allows to optimize for several discrete variables and it helps to bridge the gap between continuous and discrete variational problems.  This is of high relevance in fields different from optimal control, for example, for inverse problems the case L^p with p∈ (0,1) is of special statistical importance (e.g. for data with heavy tailed negative log probability density functions), whereas the L0 case can be the basis of a new formulation for topology optimization problems.  The techniques developed in Workpackage 1 will also be applicable to challenging problems in continuum mechanics, as for instance brittle or cohesive fracture, where singular behavior is modeled by non-smooth non-convex energies, involving truncated polynomial or L^p functionals.