Stabilising Control Laws for the Incompressible Navier-Stokes Equations using Sector Stability Theory

A method for nonlinear global stabilisation of the incompressible Navier-Stokes equations is presented and used to eliminate transient growth in linearly stable Poiseuille flow for the case of full-field actuation and sensing. In the absence of complete velocity field sensing and full actuation the controller synthesis procedure gives a controller that minimises the the attainable perturbation energy over all disturbances and thus maximises the disturbance threshold for transition to occur. The control laws are found using the theory of positive real systems, originating in the control systems community. It is found that a control law making the linearised part of the perturbed Navier-Stokes equations positive real, provides nonlinear global stability. A state-space synthesis procedure is presented that results in two game-theoretic algebraic Riccati equations.