﻿ Sum-Of-Squares Approach to Global Stability and Control of Fluid Flows - project background - nonlinear systems and SOS

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

## Nonlinear systems analysis and sum-of-squares

Stability of a dynamical system with respect to finite perturbations can be established by finding a Lyapunov function, or a Lyapunov functional in case of infinite dimensional systems. However, there is no general systematic method for constructing Lyapunov functions - the discovery of such a function is dependent on the ingenuity and creativity of the investigator.

Fortunately, a recent breakthrough in control theory (Parrilo, 2000) has provided a constructive method for generating Lyapunov functions for systems whose dynamics can be described by polynomial functions. This method is based on sum-of-squares (SOS) optimization, which reduces the problem of finding a Lyapunov function for a polynomial system to one of constructing a polynomial function that satisfies a selection of algebraic conditions. Using a number of key results from semialgebraic geometry, in particular the Positivestellensatz theorem, the resulting problem can be reformulated as an optimization problem in the form of a Semi-Definite Programme (SDP). This is particularly promising since SDP optimization problems are convex and tractable (e.g. solvable in a number of operations that is a polynomial function of the problem size), and theoretical and algorithmic research into solving such problems is extremely active. A variety of well-supported software codes for solving such problems are freely available, both for SOS problems in particular and SDP problems in general.

More recently, research has focussed on the efficient solution of semidefinite programming problems arising from SOS problems, with particular emphasis on robust optimization and exploitation of structure and sparsity in large problems. As a result of these advances, the SOS approach has found extensive applications in stability analysis, control theory and many other fields, including applications in aeronautics. The use of semidefinite programming also has a long history in systems analysis and control, since many control problems can be formulated and solved this way.

In parallel with these developments, recent work by Rantzer (2001) has provided an alternative means of demonstrating stability of a dynamic system, using a so-called dual Lyapunov or density function. Crucially, using this method it is also possible to compute simultaneously both a controller for a nonlinear system and a dual Lyapunov function certifying its stability, which is generally not possible using Lyapunov methods. For problems with polynomial dynamics, this can be achieved by solving a single convex optimization problem (again an SDP) using the SOS framework.

### Latest:

• The project has ended.
• Paolo Luchini visited us on February 2016.
• Mihailo Jovanovic and Armin Zade visited us during the two weeks 2-15 November 2015.
• All our postdocs secured positions which will allow them to contninue working in cooperation with us: Deqing Huang will be a full professor at the Southwest Jiaotong University, China, Davide Lasagna will be a Frontier Fellow at the Universiy of Southampton, UK, and Giorgio Valmorbida will move to a faculty position at the Supelec and Paris-Sud University, France. Well done!
• Next progress meeting on February 23.2015.
• Remote conference was held on 5.02.2015.
• Sergei was at UCLA attending the Mathematics of Turbulence programme. He gave two presentations about the project.
• Remote conference was held on 30.10.2014.
• Remote conference was held on 09.06.2014.
• A progress meeting was held at Imperial College on 27.03.2014.
• Charles Doering is at Imperial 25-27.03.2014 and in Oxford 27-28.03.2014 He is giving a talk on 26.03, see "Miscellaneous".
• Remote conference was held on 14.03.2014.
• Sergei gave a talk at the NeZaTeGiUs conference NeZaTeGiUs conference on 02.03.2014.
• Deqing went to China 26-30 December and gave a talk at a workshop organized by the State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou.
• Remote conference was held on 20.12.2014 The record is available from Sergei.
• Paul visited London and Oxford for two weeks in October, and took part in the project meeting on 21.10.
• Sergei gave a talk at ICNAAM 2013.
• Remote conference was held on 19.09.2013 The record is available from Sergei.
• Remore conferences were video recorded on 14.05, 06.06, 25.06, 11.07, 16.08, and 21.08.2013 The records are available for the members of the team from Sergei, but the records are not in the repository, to save space.
• July 2013. Deqing gave a talk at the School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China.
• June 5 and 6, 2013. Our remote conference was spread over 2 days this time.
• May 19-31, 2013. Sergei was in Stockholm, taking part in NORDITA Stability and Transition programme, and the follow-on SIG-33 workshop, where he gave a talk on using SoS in fluid dynamics.
• May 14, 2013. We had our first remote conference. It was interesting, and useful. Giorgio, Davide, and Deqing presented, and we all had a good chat.
• April 22, 2013. SVN repository for the project was created by Paul, hosted at ETH.
• Kick-off meeting on April 18 went fine.
• Paul is here!
• Paul Goulart will be visiting Imperial from April 15 to May 3.
• 30.03.2013 This website opens.
• 28.03.2013 Project mailing list created.
• 11-15.03.2013 Wynn visited Doering in Michigan.
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• 19.02.2013 Chernyshenko visited Southampton team.