MC 6460
Speaker
Rafael Vazquez, University of Sevilla
Title
Backstepping Boundary Control and State Estimation for Reaction-Diffusion PDEs on Arbitrary-Dimensional Balls
Abstract
In this talk, we review developments on the stabilization and state estimation of reaction-diffusion equations on n-balls (which reduce to a disk in 2-D and a sphere in 3-D) with constant coefficients, and a more recent extension with radially-varying coefficients. The backstepping method is used to design both the control law and the observer by reducing the system to an infinite sequence of 1-D systems using spherical harmonics. L^2 well-posedness and stability are proved, and in the simple case of the disk, also for higher Sobolev spaces. Remarkably, in the constant coefficient case, the resulting control law can be found explicitly. It can be written as a multiple integral whose kernel is the product of the classical backstepping kernel used in the control of one-dimensional reaction-diffusion equations and a function closely related to the Poisson kernel in the n-ball. In the radially-varying case, a new method based on power series had to be developed to prove kernel existence. Additionally, a number of further extensions, applications, and open problems will be presented.
Speaker Bio
Rafael Vazquez received the electrical engineering and mathematics degrees from the University of Seville, Seville, Spain, and the M.Sc. and Ph.D. degrees in aerospace engineering from the University of California, San Diego. He is currently Professor in the Aerospace Engineering Department and Director of the Space Surveillance Chair at the University of Seville, Spain. His research interests include control theory, estimation, and optimization with applications to distributed parameter systems, spacecraft and aircraft guidance, navigation and control, and space surveillance and awareness. He is coauthor of more than 150 journal and conference publications and the book Control of Turbulent and Magnetohydrodynamic Channel Flows. Prof. Vazquez currently serves as Associate Editor for Automatica and IEEE Control Systems Letters (L-CSS).