On the use of numerical methods for analysis and control of nonlinear convective systems

A common approach to designing feedback controllers for nonlinear partial differential equations (PDEs) is to linearize the system about an equilibrium and use the linearized model as a starting point in the design process. In many practical applications (fluid flow control, thermal fluids, etc.) the equilibrium of interest is not the trivial zero state and this equilibrium must be computed numerically. This can be a complex task, especially if the equilibrium is unstable. In an earlier paper the authors showed that even for the 1D Burgers¿ equation, standard time marching numerical schemes can produce false steady state solutions. Consequently, in this case the first step in the design and analysis process is based on an erroneous linearization and even robust controllers may fail. It has been suggested that replacing the time marching scheme by a Newton algorithm would eliminate this false equilibrium. In this short paper we illustrate that even Newton¿s method can produce a numerically false equilibrium in problems that are highly sensitivity to boundary conditions. This result has important consequences when one uses numerical tools for design and control of non-linear partial differential equations that govern typical fluid flows.