Input-to-state stability and stabilization of distributed parameter systems

控制系统的鲁棒稳定性和镇定问题是控制理论及其应用中的基础和挑战性问题。稳定性理论的里程碑之一是输入到状态稳定性理论（ISS），它是在过去二十年中为常微分方程和更一般的有限维系统而发展起来的。这个概念对于非线性系统的鲁棒稳定性分析、非线性系统的控制设计和互联系统的动力学特别有用。然而，在无限维系统领域，这个理论还远远不够完善。在现代应用中，线性和非线性分布参数系统（DPS）起着至关重要的作用。近年来，这类系统的ISS理论正变得越来越流行，但它仍然是零碎的，而且与有限维情况相比，它的发展程度要低得多。此外，在过去的十年中，已经提出了新的方法来稳定无限维系统；最值得注意的是，连续反演方法。这些方法为线性和非线性分布参数系统的鲁棒和自适应控制器的设计提供了基于iss的方法。然而，为了获得强大的控制方法，仍然需要在理解和发展这些方法方面迈出重要的一步。在这个项目中，我们将为研究分布参数系统的输入状态稳定性和稳定化建立一个坚实的基础。更具体地说，我们的目标是：1。发展线性和双线性分布参数系统的ISS理论，包括线性和双线性DPS的输入状态稳定性和稳定性准则以及ISS鲁棒性的充分条件。从有限维系统的ISS理论中得到基本非线性结果的无限维对应。特别是，ISS性质的Lyapunov表征，DPS的小增益定理和ISS在其他稳定性性质方面的表征。开发无限维系统的鲁棒镇定方法，即连续统反演的鲁棒版本，偏微分方程的有限时间鲁棒镇定以及port- hamilton系统的ISS镇定器设计。为了达到这些目标，需要具备ISS理论、泛函分析、半群理论、无限维系统理论、偏微分方程、反步设计和李亚普诺夫理论等方面的专业知识。因此，我们将作为一个团队来完成这个项目，由三个具有互补知识的小组组成，涵盖所有上述主题。该项目的目的是为ISS理论的长期发展奠定坚实的基础，使其成为广泛的非线性无限维系统的基本工具。

Robust stability and stabilization of control systems are fundamental and challenging problems in control theory and its applications. One of the milestones of stability theory is the theory of input-to-state stability (ISS), developed over the last two decades for ordinary differential equations and more general finite dimensional systems. This concept is especially useful for the analysis of robust stability of nonlinear systems, control design for nonlinear systems and the dynamics of interconnected systems.However, in the area of infinite-dimensional systems the theory is far from being complete. In modern applications, a crucial role is played by distributed parameter systems (DPS), both linear and nonlinear. ISS theory for such systems is becoming increasingly popular in recent years, but it is still fragmentary and considerably less developed than the finite dimensional case. Furthermore, over the last decade novel methods for stabilization of infinite-dimensional systems have been proposed; most notably, a continuum backstepping method. These approaches yield ISS-based methods for the design of robust and adaptive controllers for linear and nonlinear distributed parameter systems. To obtain powerful methods for control, however, major steps in theunderstanding and development of these methods are still required.In this project we are going to build a firm basis for the investigation of input-to-state stability and stabilization of distributed parameter systems. More specifically, our aims are:1. To develop an ISS theory for linear and bilinear distributed parameter systems, including criteria for input-to-state stability and stabilizability of linear and bilinear DPS and sufficient conditions for robustness of ISS.2. To obtain the infinite-dimensional counterparts of fundamental nonlinear results from ISS theory of finite-dimensional systems. In particular, Lyapunov characterizations of the ISS property, small-gain theorems for DPS and characterizations of ISS in terms of other stability properties.3. To develop methods for robust stabilization of infinite-dimensional systems, namely, a robust version of continuum backstepping, finite-time robust stabilization of partial differential equations and design of ISS stabilizers for port-Hamiltonian systems.In order to obtain these aims expertise is required in ISS theory, functional analysis, semigroup theory, infinite-dimensional systemstheory, partial differential equations, backstepping design and Lyapunov theory. Therefore we will work on this project as a team,consisting of three groups with complementary knowledge covering all of the above topics. It is the aim of the project to establish a solid basis for the long term development of ISS theory as a fundamental tool for a wide range of nonlinear infinite-dimensional systems.