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

Lead Research Organisation:
University of Oxford

Department Name: Engineering Science

### Abstract

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.

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.

### Publications

Ahmadi M
(2019)

*A framework for input-output analysis of wall-bounded shear flows*in Journal of Fluid Mechanics
Ahmadi M
(2017)

*Safety verification for distributed parameter systems using barrier functionals*in Systems & Control Letters
Ahmadi M
(2016)

*Dissipation inequalities for the analysis of a class of PDEs*in Automatica
Anderson J
(2013)

*Robust nonlinear stability and performance analysis of an F/A-18 aircraft model using sum of squares programming ROBUST NONLINEAR STABILITY AND PERFORMANCE ANALYSIS*in International Journal of Robust and Nonlinear Control
Anghel M
(2013)

*Algorithmic Construction of Lyapunov Functions for Power System Stability Analysis*in IEEE Transactions on Circuits and Systems I: Regular Papers
Chernyshenko SI
(2014)

*Polynomial sum of squares in fluid dynamics: a review with a look ahead.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Fantuzzi G
(2017)

*Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints*in IEEE Transactions on Automatic Control
Hancock E
(2013)

*Generalised absolute stability and sum of squares*in AutomaticaDescription | The aim of this project is to understand the properties of systems described by partial differential equations, such as the ones describing the motion of fluids, using computational techniques that do not need to solve these equations. A major milestone was to be able to computationally verify that certain integral inequalities hold. We have developed a method to do that and this opens the way to apply this method to a number of problems related to dynamical systems in areas such as fluid mechanics, chemical reactions and heat transfer, all described by partial differential equations. This method is being used by other academics to answer questions about their models, without the need to simulate them. |

Exploitation Route | By using the software package we developed (intsostools, https://github.com/gvalmorbida/INTSOSTOOLS) and also using our mathematical results. |

Sectors | Environment |

URL | https://www.imperial.ac.uk/aeronautics/fluiddynamics/sumofsquares/index.php |

Description | The project's aim is to find a way to understand mathematical models of physical systems described by Partial Differential Equations (PDEs) without solving these PDEs. The results we obtained are theoretical and mathematical in nature but we expect that in the future they will be used to result in societal and economical impact. Especially since the final publication of the JFM paper, we expect that industry will be interested in adopting some of our techniques to provide provable bounds on properties of their designs. |

First Year Of Impact | 2019 |

Sector | Environment |

Description | Collaboration on Sum of Squares and Fluid Mechanics with UCSB |

Organisation | University of California, Santa Barbara |

Country | United States |

Sector | Academic/University |

PI Contribution | Collaboration, exchange of ideas and knowledge |

Collaborator Contribution | Collaboration, exchange of ideas and knowledge |

Impact | Ongoing. |

Start Year | 2013 |