Order preserving operators in problems of optimal control and in the theory of partial differential equations

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Positivity and order structures in (normed) vector spaces as well as positive and structure preserving operators on these spaces play an important role in operator theory and its applications such as dynamical systems and partial differential equations. In this project we propose to study, together with our partners from Moscow, Voronezh and Vladikavkaz, optimal control problems in ordered spaces, order structures and order preserving operators and semigroups of operators on partially ordered, normed spaces which are in general not Riesz spaces. This project proposal replies to a call about a DFG-RFBR Cooperation. This project proposal is also a continuation of an Initiation of an International Collaboration (from July 2014 to June 2015, together with members of the Russian Academy of Sciences at Vladikavkaz). A previous version of this project proposal has been submitted in the framework of a DFG-RSF Cooperation in September 2015 (GZ: CH 1282/3-1) and was evaluated positively on the German side, but not on the Russian side. We still intend to deepen this collaboration and expect from this international cooperation new insights and results for linear and nonlinear order preserving operators on pre-Riesz spaces, finite elements in spaces of such operators as well as for positive operator semigroups on ordered Banach spaces.