Backstepping Boundary Control of Navier-Stokes Channel Flow: Explicit Gain Formulae in 3D

In a companion ACC ´06 paper we present a boundary control law, designed using the backstepping method, that stabilizes the 3D Navier-Stokes system linearized around the much studied Poiseuille flow profile. This control law employs three controllers that actuate solely along one boundary. The controller that acts in the wall normal direction prepares the system for the backstepping method, while the other two controllers, which act in the streamwise and spanwise direction, are designed using backstepping to stabilize the system. They do so by decoupling the normal vorticity from the normal velocity and then stabilizing the normal velocity. Instead of solving Ricatti equations to find the kernels for the controller gain, the kernels are found by solving certain linear hyperbolic pdes offline. We study the pdes derived in the companion paper that define the gain kernels. We focus first on the specific case of perturbations with no streamwise dependence. This is equivalent to setting the streamwise wavenumber to zero. We derive explicit solutions to the gain kernels for this important case where transient growth is the largest. In addition, we solve a series of related pdes in order to find an approximate solution to the gain kernel pdes when the streamwise and spanwise wavenumbers are small