We point to selected research highlights which could be achieved during the grant period. Non-smooth sparsity-promoting functionals were analyzed in the context of optimal stabilization problems. The peculiarity of this work is that it could be verified that the optimal controls completely shut down once they are sufficiently close to a stable steady state. In our work on nonconvex penalization of switching control of partial differential equations, optimal control problems with switching constraints, were analysed. New, primal-dual based, numerical schemes for non-convex optimization problems were proposed and analyzed for image reconstruction and fracture mechanics.
The synthesis of optimal feedback policies in the context of non-smooth/non-convex optimization framework was investigated. The relevance of this work is that it is one of the first attempts to understand the interplay of nonsmooth costs with feedback controls and Hamilton-Jacobi-Bellman (HJB) equations. In particular, it presents a characterization of parsimonious control actions in terms of sparsity, and switching conrtols.
Another application driven topic was the study of real-time synthesis for multiscale agent-based models originating from social dynamics and crowd motion by model predictive control techniques. The design of model predictive control strategies to reduce human congestion in evacuation scenarios was developed. Model predictive control techniques are essential to reduce the associated computational burden and to achieve real-time computability. We validated their use for diverse classes of PDE-constrained optimal control problems. Viscosity methods for large deviations estimates of multiscale stochastic processes for Hamilton-Jacobi-Bellman (HJB) equations were studied with special attention to the financial applications. Nonlocal HJB equations related to discontinuous stochastic processes were analyzed as well.
For the monodomain equation, arising in context of electro-cardiology modelling, lack of exact null-controllability could be established. For the one-dimensional case, however, together with moving controllers, exact null-controllability was verified, and optimal boundary stabilizability results were obtained for the 3-D case case. Further, stabilizability by finite dimensional controls was investigated for this class of equations.In a series of highly technical papers, Taylor expansions for the value function associated to optimal control problems were developed, and their numerical relevance was investigated. Several concepts to weaken the curse of dimensionality when computing optimal feedback gains were proposed and analyzed. They rest, in part, on approximating HJB equations, to the other part on machine learning techniques.
