Interconnected infinite-dimensional systems for heterogeneous media

受机械、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，无限维系统的分析和控制在过去几十年中成为一个主要的兴趣领域。经典系统理论的基本概念已经逐步推广到无限维系统，数学界和工程界都做出了贡献。最近，科学家们开始对理解由分布参数子系统（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。大多数现有的文献建模，系统分析和控制，处理网络的均匀系统，如桁架的弹性杆或导热性能的金属泡沫。在这个项目中，我们将考虑网络的异构系统，这也涉及网络中的换能器提供驱动，例如在智能泡沫声学，或耦合质量和热传输现象的催化泡沫。我们将开发用于分析、模型简化、仿真和控制的工具。本研究将经典的无穷维系统分析方法与端口-哈密顿系统理论相结合，重点研究一类描述流体-结构-热相互作用的非均质系统。该项目的目的是开发工具的分析和控制这些系统使用控制理论工具，如理论的良好，本文将结合Hamilton系统的无源性和几何性质，讨论非线性反馈下的非线性系统及其鲁棒性，另一个重要的课题将是通过保持偏微分方程的一些控制理论性质或互连接口。该项目旨在为PH系统开发保持几何结构的简化方法，无论是泊松括号还是狄拉克结构。例如，在图或更一般的k-复合体上定义的PDE系统的情况下，这可以导致计算工具获得在具有期望物理性质的聚类图或k-复合体上定义的简化模型。这些PDE的异构系统的控制将通过开发特定的工具来解决，例如通过设计封闭的循环使用与具有指定互连结构和物理（例如热力学）性质的PH系统的等效性。此外，模型的PH结构，在有限和无限维，应用于开发适应的观测器和计算方法的轨迹规划。与合作伙伴的专业领域相对应的测试台将作为演示。

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.