Interconnected infinite-dimensional systems for heterogeneous media

在机械、航空、能源系统和化学工程以及新型计算工具的最新技术进步的推动下，无限维系统的分析和控制在过去几十年中成为一个主要感兴趣的领域。经典系统理论的基本概念在数学和工程界的贡献下，已经逐步推广到无限维系统。最近，科学家开始对理解分布参数子系统（由偏微分方程系统描述）组成的系统感兴趣，这些子系统在网络中相互作用。现有的大多数关于建模、系统分析和控制的文献都涉及均匀系统网络，如弹性杆桁架或金属泡沫的导热性。在这个项目中，我们将考虑异质系统的网络，这也涉及网络中的传感器来提供驱动，例如用于声学的智能泡沫，或催化泡沫中的耦合质量和热传递现象。我们将为它们的分析、模型简化、仿真和控制开发工具。在这个项目中使用的方法，结合了无限维系统的经典工具和互连端口-哈密顿（PH）系统的理论。我们集中研究的一类非均相系统是那些描述流-固-热相互作用的系统。该项目旨在开发用于分析和控制这些系统的工具，使用控制理论工具，如适定系统理论及其对非线性反馈的鲁棒性，结合无源性和哈密顿系统的几何特性。另一个重要的课题将是通过保留pde的一些控制理论性质或互连接口来开发这种异构系统的模型简化技术。该项目旨在开发还原PH系统的方法，以保持几何结构，无论是泊松支架或狄拉克结构。这可能导致例如在图上定义的偏微分方程系统或更一般的k-复合体的情况下，使用计算工具来获得在聚类图或具有所需物理性质的k-复合体上定义的简化模型。最后，将通过开发特定的工具来解决这些异构pde系统的控制问题，例如PH系统，通过设计与具有特定互连结构和物理（例如热力学）性质的PH系统等效的闭环。此外，模型的PH结构，在有限和无限维，应使用发展适应的观测器和计算方法的轨迹规划。试验台与合作伙伴的专业领域相对应，将作为示范。

Motivated by recent technological progress in mechanical, aeronautical, energy systems and chemical engineering and novel computational tools, the analysis and control of infinite-dimensional systems became a field of major interest during the last decades. The basic concepts of classical systems theory have been progressively generalized to infinite-dimensional systems with contributions stemming from the mathematical as well as from the engineering community.More recently, scientists became interested in understanding systems composed of distributed-parameter subsystems (described by systems of partial differential equations, PDEs) which interact in networks. Most of the existing literature on modelling, system analysis and control, deals with networks of homogeneous systems such as trusses of elastic rods or the heat conductivity properties of metal foams. In this project, we shall consider networks of heterogeneous systems, which also involves transducers in the network to provide actuation, as for instance in smart foams for acoustics, or coupled mass and heat transport phenomena in catalytic foams. We will develop tools for their analysis, model reduction, simulation and control. The methodology used in this project, combines classical tools for infinite-dimensional systems and the theory of interconnected port-Hamiltonian (PH) systems.A class of heterogeneous systems that we aim to study intensively are those describing fluid-structure-thermal interactions. The project aims to develop tools for the analysis and the control of these systems using control theoretic tools such as the theory of well-posed systems and its robustness with respect to nonlinear feedbacks in combination with passivity properties and the geometry of Hamiltonian systems.Another important topic will be the development of model reduction techniques for such heterogeneous systems of PDEs by preserving some of their control theoretic properties or interconnection interfaces. The project aims at developing reduction methods for PH systems that preserve geometric structures, either Poisson bracket or Dirac structure. This may lead for instance in the case of systems of PDEs defined on graphs or more general k-complexes, to computational tools to obtain a reduced model defined on a clustered graphs or k-complexes with desired physical properties.Finally, the control of these heterogeneous systems of PDEs will be addressed by developing specific tools for instance for PH systems by designing the closed-loop using equivalence with PH systems with specified interconnection structure and physical (e.g. thermodynamic) properties. Moreover, the PH structure of the models, in finite and in infinite dimension, shall be used to develop adapted observers and computational methods for the trajectory planning. Test benches corresponding to the field of expertise of the partners, will serve as demonstrators.