Stabilization of Nonlinear PDE's and Applications to Control of Flows

摘要

We have contributed in three general areas that are relevant to high performance control of aerospace vehicles. We have developed systematic techniques for real time optimization using the method of extremum seeking. This method is now highly general applicable to multivariable problems, to problems with time-varying parameters, to problems involving slow dynamics where fast convergence is demanded, to problems where the objective is convergence to any value of the gradient (not just zero). to problems that involve limit cycles (unsteady extrema). and to problems that evolve in discrete time. Examples of applications that we have pursued include formation flight, compressor stall and surge, arid thermoacoustic combustion instabilities. In 2003 we published a research monograph on extremum seeking. In the area of flow control we have pioneered the use of stabilization for drag reduction and mixing. Employing Lyapunov techniques, we have solved flow control problems in channels, pipe geometries, flows around bluff bodies, and jet flows. In 2002 we published the first book dedicated to control algorithm design for fluid flows. For a broad class of linear parabolic distributed parameter systems we pioneered a back- stepping method for solving the boundary control stabilization problems. This is the first method that yields explicit solutions for both the control laws and for the closed-loop solutions. The implications of this are impossible to overstate-the design is direct and free of numerical issues, and the well posedness analysis is trivial (it is a bonus to the explicit design process). For several nonlinear PDEs we solved global stabilization problems in the presence of parametric uncertainties and input dynamics, paving the way for a future general theoretical development for nonlinear PDEs.