Stabilization of linear control systems with parameter-dependent system matrices is investigated. A Riccati based feedback mechanism is proposed and analyzed. It is constructed by means of an ensemble of parameters from a training set. This single feedback stabilizes all systems of the training set and also systems in its vicinity. Moreover its suboptimality with respect to optimal feedback for each single parameter from the training set can be quantified. This is joint work with P. Guth and S. Rodrigues

We consider optimal control problems with random coefficients where the deterministic control should optimize a (now random) objective functional in a suitable sense. Minimizing the expectation of the objective over the random input leads to a "risk-neutral" stochastic optimization problem, which can undervalue extreme (rare but expensive) events. Instead, we consider a risk-averse formulation where the expectation is replaced by a convex risk measure that is more sensitive to extreme events such as the Conditional Value-at-Risk (CVar). We show that such problems can be solved with a nonlinear primal-dual proximal splitting method and how the solution can be accelerated by a stochastic approach where in every iteration only a "minibatch" of scenarios is evaluated.

A number of works have been devoted to the variational analysis of so-called spectral functions, most of which focus on specific settings, such as the spaces of symmetric/rectangular matrices or Euclidean Jordan algebras. In this talk, we introduce an abstract framework of "spectral decomposition systems," which unifies a broad range of previously studied settings and facilitates the derivation of new results in more general contexts, including normal decomposition systems. Within this framework, we characterize various variational-analytic objects and properties associated with spectral functions — such as convexity, Legendre-Fenchel conjugate, (generalized) subdifferentials, and Bregman proximity operators — via those of the corresponding reduced functions. Joint work with Hoa Bui (Curtin University) and Christian Clason (Universität Graz).

An important class of nonlinear PDEs can be formulated as variational problems on nonlinear spaces, such as Hilbert manifolds. To solve these problems efficiently, also in the presence of non-smoothness, a semi-smooth Newton method can be applied. In this talk we will extend the Newton's method to mappings between nonlinear manifolds and present a geometric version of this concept for the case of variational problems, which can be formulated via mappings from a manifold into a dual vector bundle. To render Newton steps well defined in this setting, we have to equip the vector bundle with a connection. We will discuss local convergence and provide some numerical examples.