Sum-of-Squares Approach to Global Stability and Control of Fluid Flows

本项目旨在发展分析流体流动稳定性和流动控制的新方法。气流控制是最有希望减少阻力的途径之一，从而减少碳排放，这是当今航空业面临的最大挑战。然而，流体流动的稳定性分析提出了重大的数学和计算挑战。该项目是基于最近在多项式的正确定性相关的数学重大突破。正确定性在稳定性和控制理论中很重要，因为它是李雅普诺夫函数的一个基本性质，它是建立给定系统稳定性的有力工具。自1892年李雅普诺夫函数被引入以来，一个多世纪以来，构造李雅普诺夫函数一直依赖于研究人员的聪明才智和创造力。2000年，人们发现了一种系统的、数值上易于处理的方法来构造平方和多项式，并满足一组线性约束。如果一个多项式是其他多项式的平方和，那么它就是正定的。因此，系统的、计算机辅助的李雅普诺夫函数的构造对于用多项式非线性方程描述的系统成为可能。在过去的十年中，平方和方法在几个研究领域得到了广泛的应用，并产生了重大影响。控制不可压缩流体运动的Navier-Stokes方程具有多项式非线性。该项目将通过应用平方和方法来实现由这些方程控制的流体流动的稳定性和控制，从而实现其目标。这将需要发展新的先进的分析技术，并结合广泛的数值计算。该项目具有基础性质，其主要预期结果适用于各种流体流动。旋转的泰勒-库埃特流将是所开发方法应用的第一个对象。Taylor-Couette流在广泛的工业应用中，由于各种原因在稳定性理论中具有标志性的地位，传统上作为新方法的试验台。为了最大限度地发挥研究的影响，除了传统的期刊论文和会议传播途径外，项目合作者将以非正式和正式讲习班的形式为工业界和学术界开展有针对性的传播活动。业界的代表将获邀出席工作坊。更广泛的受众将通过一个专门创建和持续维护的网页。

This project aims at developing new methods of analysis of the stability of fluid flows and flow control. Flow control is among the most promising routes for reducing drag, thus reducing carbon emissions, which is the strongest challenge for aviation today. However, the stability analysis of fluid flows poses significant mathematical and computational challenges. The project is based on a recent major breakthrough in mathematics related to positive-definiteness of polynomials. Positive-definiteness is important in stability and control theory because it is an essential property of a Lyapunov function, which is a powerful tool for establishing stability of a given system. For more than a century since their introduction in 1892 constructing Lyapunov functions was dependent on ingenuity and creativity of the researcher. In 2000 a systematic and numerically tractable way of constructing polynomials that are sums of squares and that satisfy a set of linear constraints was discovered. If a polynomial is a sum of squares of other polynomials then it is positive-definite. Thus, systematic, computer-aided construction of Lyapunov functions became possible for systems described by equations with polynomial non-linearity. In the last decade the Sum-of-Squares approach became widely used with significant impact in several research areas.The Navier-Stokes equations governing motion of incompressible fluid have a polynomial nonlinearity. This project will achieve its goals by applying sum-of-squares approach to stability and control of the fluid flows governed by these equations. This will require development of new advanced analytical techniques combined with extensive numerical calculations. The project has a fundamental nature, with main expected outcomes being applicable to a large variety of fluid flows. The rotating Taylor-Couette flow will be the first object to which the developed methods will be applied. Taylor-Couette flow, encountered in a wide range of industrial application, for a variety of reasons has an iconic status in the stability theory, traditionally serving as a test-bench for new methods.In order to maximise the impact of the research, the project collaborators will conduct targeted dissemination activities for industry and academia in the form of informal and formal workshops, in addition to traditional dissemination routes of journal papers and conferences. Selected representatives from industry will be invited to attend the workshops. Wider audience will be reached via a specially created and continuously maintained web page.