Systems theory of partial differential-algebraic equations

许多与输入、状态和输出变量具有实际相关性的动态系统被建模为偏微分方程组和/或微分-代数方程组，例如电路或经典机械系统。在该项目中，具有无限维状态空间的微分-代数系统被用来描述这类系统。虽然有限维和无限维常微分方程组的系统理论已经相当成熟，但对于有限维和无限维系统，微分-代数系统仍然是具有挑战性的公开问题。例如，在模型预测控制中忽略了微分-代数方程的结构性质，将其视为最优控制问题中的另一个约束。我们的目标是促进(部分)微分-代数方程理论的发展。特别是，我们的目标是获得关于结构属性的基本见解，如零动态和所谓的外部传递函数。然后，我们想利用我们在有限和无限维微分-代数系统的最优和模型预测控制方面的研究成果。为此，功能分析方法以及时变系统理论有望成为我们方法的关键组成部分。

Numerous dynamical systems of practical relevance with input, state, and output variables are modelled as systems of partial differential equations and/or systems of differential-algebraic equations, e.g. electrical circuits or classical mechanical systems. Within the proposed project, differential-algebraic systems with an infinite dimensional state space are used in order to describe this class of systems.While systems theory is already quite mature for finite and infinite dimensional systems of ordinary differential equations, there are still challenging open problems for differential-algebraic systems both for finite and infinite dimensional systems. For example, structural properties of differential-algebraic equations are ignored in model predictive control where the algebraic equations are considered as another constraint in the optimal control problem.Our goal in the proposed project is to contribute to the development of the theory of (partial) differential-algebraic equations. In particular, we aim at gaining fundamental insight with respect to structural properties like zero dynamics and so called outer transfer functions. Then, we want to make use of our findings in optimal and model predictive control of finite and infinite dimensional differential-algebraic systems. To this end, functional analytic methods as well as the theory of time-varying systems are expected to serve as key ingredients in our approach.