Global stability analysis of fluid flows using sum-of-squares

P. J. Goulart and S. Chernyshenko

Physica D, vol. 241, no. 6, pp. 692-704, March 2012.
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@article{GC:2012,
  author = {P. J. Goulart and S. Chernyshenko},
  title = {Global stability analysis of fluid flows using sum-of-squares},
  journal = {Physica D},
  year = {2012},
  volume = {241},
  number = {6},
  pages = {692-704},
  url = {http://dx.doi.org/10.1016/j.physd.2011.12.008},
  doi = {10.1016/j.physd.2011.12.008}
}

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite dimensional Navier-Stokes systems using robust optimization techniques. Crucially, this extension requires only the solution of infinite-dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization problems. We further show that subject to minor technical constraints, a general polynomial Lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower-bound on the maximum Reynolds number for which a flow is globally stable, if the flow does remain globally stable for Reynolds numbers at least slightly beyond the energy stability limit. Such polynomial functions can be searched for efficiently using the SOS technique we propose.