Sensitivity analysis of turbulence using unstable periodic orbits: a demonstration on the Kuramoto-Sivashinsky equation

A robust approach for adjoint-based sensitivity analysis of chaotic dynamics based on unstable periodic orbits is proposed. We show that a careful reformulation of established variational techniques to such trajectories enables the sensitivity of time averages with respect to design parameters to be calculated exactly, regardless of the stability characteristics and length of the orbit. This holds the promise of bringing recent advances in the study of the dynamics and role of exact solutions of the Navier-Stokes equations to bear in design and optimisation problems for turbulent flows. In this paper, we derive the adjoint technique and discuss, as a proof of concept, a feedback control design problem for the Kuramoto-Sivashinsky equation, i.e. a prototypical onedimensional partial differential equation with rich dynamical behaviour. Key challenges and opportunities associated to the application of this method to fluid turbulence, left as future work, are also discussed.