Viscosity solutions to systems of variational inequalities related to multi-modes switching problems
Many production problems can be phrased as finding an optimal management strategy
Many production problems can be phrased as finding an optimal management strategy for a production facility that can operate in several different modes, leading to so-called multi-modes switching problems that are studied as systems of partial differential equations.
Such problems have been intensively studied over the last decades from the perspective of viscosity solutions. Unfortunately, however, these impressive achievements have thus far stayed largely confined to the mathematical real with only little impact on real-world problems. One reason why multi-modes switching theory have not been more widely applied is that the theory is mainly developed for unbounded spatial domains while many practical problems require results for bounded spatial domains. Building on the famous theory of viscosity solutions we intend to overcome this lack of theory by proving fundamental properties such as existence, uniqueness and regularity of viscosity solutions to these kind of systems in bounded spatial domains equipped with suitable boundary conditions and geometric assumptions. To strengthen the important bridge between mathematical results and real world applications, we aim to apply existing as well as the anticipated mathematical results to derive efficient management strategies for energy production and maintenance, e.g. in the setting of run-of river hydropower.