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

赋范向量空间中的正性和序结构以及这些空间上的正性和保结构算子在算子理论及其应用（如动力系统和偏微分方程）中起着重要的作用。在这个项目中，我们建议研究，连同我们的合作伙伴从莫斯科，沃罗涅日和弗拉迪卡夫卡兹，最优控制问题的有序空间，秩序结构和秩序保持运营商和半群的运营商偏序，赋范空间一般不Riesz空间。本项目建议书是对DFG-RFBR合作呼吁的回应。该项目提案也是国际合作启动的延续（从2014年7月至2015年6月，与弗拉迪卡夫卡兹的俄罗斯科学院成员一起）。2015年9月，在DFG-RSF合作框架内提交了该项目提案的前一版本（GZ：CH 1282/3-1），并在德国方面得到了积极评价，但在俄罗斯方面没有。我们仍然打算深化这种合作，并期望从这种国际合作的新的见解和结果的线性和非线性保序算子的pre-Riesz空间，有限元素的空间，这样的运营商以及正算子半群有序Banach空间。

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.