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

控制系统的鲁棒稳定性和镇定是控制理论及其应用中的基础性和挑战性问题。稳定性理论的里程碑之一是输入到状态的稳定性(ISS)理论，它是近二十年来发展起来的常微分方程组和更一般的有限维系统。这一概念对于分析非线性系统的鲁棒稳定性、非线性系统的控制设计以及关联系统的动力学都是非常有用的，但在无穷维系统领域，这一理论还很不完善。在现代应用中，分布参数系统(DPS)扮演着重要的角色，包括线性和非线性。这类系统的ISS理论在最近几年变得越来越流行，但它仍然是支离破碎的，而且比有限维情况下的发展要差得多。此外，在过去的十年里，已经提出了用于无穷维系统镇定的新方法，其中最著名的是连续统反推方法。这些方法产生了基于ISS的线性和非线性分布参数系统的鲁棒自适应控制器设计方法。然而，要获得强有力的控制方法，仍需要在理解和发展这些方法方面采取主要步骤。在这个项目中，我们将为研究分布参数系统的输入-状态稳定性和镇定奠定坚实的基础。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，包括线性和双线性分布参数系统输入-状态稳定性和可镇定的判据以及ISS的稳健性充分条件。从有限维系统的ISS理论中得到基本非线性结果的无穷维对应项。特别地，给出了ISS性质的Lyapunov刻画、DPS的小增益定理以及ISS的其他稳定性刻画。为了发展无穷维系统的鲁棒镇定方法，即连续介质反推的鲁棒形式、偏微分方程的有限时间鲁棒镇定和端口-哈密顿系统的ISS稳定器的设计，需要ISS理论、泛函分析、半群理论、无穷维系统理论、偏微分方程反推设计和Lyapunov理论方面的专业知识。因此，我们将作为一个团队工作在这个项目上，由三个小组组成，他们拥有涵盖所有上述主题的互补知识。该项目的目的是为ISS理论的长期发展奠定坚实的基础，ISS理论是一种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.