For a vast variety of fluid flows the dynamics are governed by the Navier-Stokes equations which are highly nonlinear. In particular, the corresponding Galerkin models involve a quadratic type nonlinearity. The latter incudes the Burgers' equation as well.In this paper, a computational algorithm for nonlinear balanced truncation of the Galerkin models is proposed. The method is based on Taylor series expansion in which the computation of successively higher order terms reduces to solving consecutively higher order systems of linear algebraic equations. This procedure is also used to compute the coordinate transformation that yields the balanced system. Formulas to compute the nonlinear controllability, observability functions and coordinate transformation are derived.

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In this paper, the popular proper orthogonal decomposition (POD) without the usual integral or inner product constraints is extended to general Hilbert spaces, such as Sobolev spaces, using functional analytic methods. It is shown that a particular tensor product space is dense in the Hilbert space where the partial differential equation (PDE) solution lives. This allows approximating the PDE solution by tensors to any desired accuracy. Optimal approximation by these tensors is shown to result in the POD using operator theoretic arguments. This is achieved by solving a nonlinear optimization problem where the PDE solution is approximated by operators of a prescribed finite rank in the corresponding trace class 2 norm. POD modes can then be computed by solving an infinite dimensional eigenvalue problem using Hilbert-Schmidt theory. Moreover, an optimal method in constructing reduced order models for the two-dimensional Burgers' equation subject to boundary control is presented and compared to the POD reduced models. A closed-loop feedback controller then designed using the reduced order model and then applied to the full order model.