Microscopic simulations and nonlinear control of dissipative distributed processes

A nonlinear continuous-time feedback controller design methodology is developed for distributed processes, whose dynamic behavior can be described by microscopic evolution rules. Employing the micro-Galerkin method to bridge the gap between the microscopic-level evolution rules and the "coarse" process behavior, "coarse" process steady states are estimated and nonlinear process models are identified off-line through the solution of a series of nonlinear programs. Subsequently, nonlinear output feedback controllers are designed, on the basis of the identified process model that enforce stability in the closed-loop system. The method is used to control a system of coupled nonlinear one-dimensional PDEs (the FitzHugh-Nagumo equations), widely used to describe the formation of patterns in reacting and biological systems. Employing kinetic theory based microscopic realizations of the process, the method is used to design output feedback controllers that stabilize the FHN at an unstable, nonuniform in space, steady state.