Carlos N. Rautenberg

A wide range of problems in applied sciences involve constraints on variables of interest. These naturally arise in modeling of complex physical phenomena but also appear as a result of hierarchy or competition. Two different classes of these constraints can be described: explicit, where the bounds are known in advance, and implicit, where the bounds depend on the solution of the problem itself. One simple example of an implicitly constrained problem is that of finding the position of an elastic membrane with an obstacle that deforms upon the action of the membrane. In this example, the membrane position is the variable of interest, and the position of the obstacle is the implicit bound or constraint. The control and parameter identification for this class of implicitly constrained problems represent a significant challenge for a large variety of problems. Some possible applications include the design of composite materials that sustain large forces without plastic deformation, the manufacture of multilayer organic light emitting diodes (OLEDs), and the detection of subsurface cracks in buildings that may compromise structural integrity and lead to catastrophic failure.

Motivated by efficient energy distribution, we develop theory and solution algorithms for a new class of generalized Nash equilib- rium problems (GNEPs) arising in game theoretic formulations of energy markets. As agents of the GNEP measure data along the underlying equilibrium process, we establish model predic- tive control (MPC) and closed loop strategies to target realistic control scenarios. The pertinent transport physics together with additional system constraints are considered within a hierarchy of models and with possible stochastic perturbations.

The optimal sensor placement problem for the estimation of the temperature distribution in buildings is a highly nonlinear and multi-scale problem where stochastic perturbations are usually present. The main goal here is to properly locate sensors in order to reliably estimate the temperature distribution in certain areas. Since feedback controllers are usually in use, a proper estimation of the state is of utmost importance in order to reduce energy consumption of such control systems.

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Quasi-variational inequalities (QVls) often arise in applications where non-smooth and nonlinear phenomena lead to complex state-dependent constraints. This project is devoted to analyzing and numerically solving optimal control problems associated with elliptic and parabolic QVIs. In such problems, QVIs constitute non-linear and non-smooth constraints and enhance theoretical and numerical difficulties.

Quasi-variational inequalities (QVIs) were introduced by Bensoussan and Lions, and are generalizations of variational inequalities (VIs) where the associated constraint set is not known a priori. A rigorous mathematical framework on optimal uncertainty quantification that is suitable for this type of problems, as well as their risk averse treatment is lacking.

We pursue first steps for deriving feedback laws for nonsmooth optimal control problems, in particular involving moving boundaries in a sharp interface setting. Such feedback control allows for automation including dynamic integration of measurements into optimization strategies.