Control of Turbulent and Magnetohydrodynamic Channel Flows

Preface

This monograph presents new constructive design methods for boundary stabi-lization and boundary estimation for several classes of benchmark problems for flow control, with potential applications in turbulence control, weather forecast- ing, and plasma control. 

The basis of our approach is the recently developed continuous backstepping method for parabolic PDEs [90]. We expand the appli-cability of boundary controllers for flow systems from low Reynolds number  to high Reynolds number conditions.

Efforts in flow control over the last few years have led to a wide range of develop- ments in many different directions, reflecting the interdisciplinary character of the research community (composed of control theorists, specialists in fluid me- chanics, mathematicians, and physicists). However, most implementable devel- opments so far have been obtained using discretized versions of the plant mod- els and finite-dimensional control techniques. In contrast, our design method is based on the “continuum” version of the backstepping approach, applied to the PDE model of the flow. 

The postponement of the spatial discretization until the implementation stage offers advantages that range from numerical to analytical. In fact, our methods offer a rather unparalleled physical intuition by forcing the closed-loop systems to dynamically behave as well-damped heat equation PDEs.

This constructive design philosophy is particularly rewarded in terms of the ob- server and control gains that we derive. In all cases these gains are presented as explicitly computable formulas such as rapidly convergent symbolic recursions, solutions to well-posed solvable linear PDEs, or even directly as closed-form analytical expressions. 

For all designs we state and prove mathematical results guaranteeing closed-loop stability and observer convergence. This constructive approach has allowed us to obtain the first nontrivial closed-loop explicit solu- tion for the 2D Navier–Stokes channel flow model.