Boundary Control of PDEs and Applications to Turbulent Flows and Flexible Structures

We present a tutorial overview of new methods for designing feedback controllers for infinite-dimensional systems controlled from the boundary. Such problems arise in turbulent flow control, chemical process control, fusion, nanotechnology applications such as control of cantilever beams in atomic force microscopes, etc. Due to the inherent high dimensionality and unbounded input and output operators, control methods that mimic standard finite-dimensional techniques and require the solution of operator Riccati equations are hard to apply to these problems. We present a set of novel controller and observer design techniques inspired by geometric nonlinear control tools and "backstepping," which employ only symbolic computation. In flow control applications these techniques can be employed in such a way that not only the turbulence is suppressed but the linearized Navier-Stokes PDEs are solved explicitly. In control of flexible beams the new techniques remove the need for collocated actuators and sensors, which has permeated this field for decades. For distributed parameter systems with unknown parameters we show adaptive control techniques that ensure stabilization even in the most challenging of cases like the cases with infinite relative degrees, infinitely many unknown parameters, and infinitely many unmeasured states. Trajectory tracking problems are rarely, if ever, pursued for PDE systems. We show that they become readily tractable with the new techniques. All of the new techniques are fully accessible to researchers with an engineer\´s level of mathematical background.