Boundary control of the Navier–Stokes equation by empirical reduction of modes

The Karhunen–Loeve Galerkin procedure is a type of Galerkin methods that employs the empirical eigenfunctions of the Karhunen–Loeve decomposition as basis functions. This technique can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena. Previously [1] , it has been shown that one dimensional Burgers equation is reduced to a low dimensional model by this method, which is employed to solve boundary optimal control problems very efficiently. The present paper demonstrates that the Karhunen–Loeve Galerkin procedure can be extended to solve problems of the boundary optimal control of multidimensional Navier–Stokes equations. Since the reduction of modes in the multidimensional case is much larger than that in the one dimensional case, the present technique is found to be more powerful when applied to the control problems of the Navier–Stokes equation than those of the Burgers equation.