Infinite-dimensional backstepping and applications to flows in electromagnetic fields

Numerous physical processes are modeled by partial differential equations and are often instrumented with boundary actuators and sensors. A major new effort has been underway in recent years to develop constructive designs of boundary control laws for unstable PDE systems. This development draws upon the ideas of "backstepping" synthesis for nonlinear ODEs from the 1990s and is an infinite-dimensional, continuum extension of backstepping. Initial efforts on infinite-dimensional backstepping focused on linear PDEs and produced a set of methodologies, for all of the major classes of PDEs, that results in elegant formulae for the gain functions of the feedback laws, which do not require the solution of operator Riccati equations. The most recent efforts pushed even further, into developing an adaptive control approach for PDEs, with unknown functional parameters and output feedback, and into developing feedback linearizing control designs for nonlinear PDEs. The most significant application-driven results in this expanding area of research have so far been for turbulent and magnetohydrodynamic flows, such as those that arise in aerodynamics applications, and in plasma and liquid metal flow problems in tokamak fusion reactors. This talk will present highlights of various methods and applications of infinite-dimensional backstepping.