| The dissertation introduces a constructive and rigorous approach to design boundary controllers and boundary estimators for flow control problems.; We study several flow control applications, including thermal fluids with convective-type instabilities, Navier-Stokes channel flow, and magnetohydrodynamic channel flow, which are considered benchmark models to problems such as turbulence control, drag reduction, model-based turbulence estimation, cooling systems (computer systems, fusion reactors), hypersonic flight and propulsion. For all systems considered, we solve both the problem of (full state) boundary stabilization and boundary estimation, and the combination of the two, output feedback boundary stabilization. We also consider a tracking problem for the Navier-Stokes channel flow.; While some of these problems have been solved in the past, previous solutions were mainly done for spatially discretized versions of the models; however, controllers designed for a discretized plant do not always converge (when the grid size goes to zero) to stabilizing controllers for the continuous plant. In addition, the computational complexity of discretization-based methods may become overwhelming if a very fine grid is necessary in the discretizations to accurately describe the system (for example, in the case of very large Reynolds numbers in the channel flow stabilization problem).; In contrast, our method uses a continuum approach and does not require discretization. Our approach exploits the spatially invariant geometry of our model problems to apply the backstepping control/observer design method for 1-D infinite-dimensional linear parabolic systems. Backstepping produces explicitly computable control and output injection gains, which are found from the solutions of linear hyperbolic PDEs. For all considered problems we show stability of the closed-loop system (for stabilization problems) and observer estimate convergence (for estimation problems).; Finally, we also provide an extension of the backstepping method that allows to solve boundary control problems for a class of nonlinear parabolic PDEs. While boundary control of linear parabolic PDEs is a well established subject, boundary control of nonlinear parabolic PDEs is still an open problem as far as general classes of systems are concerned. Applications of interest include not only fluids but also many others like flexible structures, atomic force microscopy, aeroelasticity, chemical systems and quantum systems. |
