On Smoothing Properties of PDEs and Abstract Controllability Results in Hilbert Space with Applications to Boundary Control of Systems governed by PDEs.
by
Steve Taylor
University of Auckland
Coauthors: Walter Littman (University of Minnesota)

Most people who have worked with PDEs are probably familiar with the smoothing nature of parabolic PDEs, such as the heat equation. The physical interpretation is that an initially rough temperature distribution becomes smoother as time increases (in fact, for the heat equation with constant coefficients, the temperature is analytic for all t > 0). However, PDEs that can also be solved in the direction of decreasing time (i.e. to see what happened in the past), such as the wave equation and the Schrödinger equation, cannot have such global smoothing properties. I will briefly explain why this is the case and I will also illustrate with some examples that such equations may still have local smoothing properties.

Next, I will discuss some abstract results about evolution operators in a Hilbert space setting which make use of this type of smoothing property. I will show how the results can be used to control solutions of PDE systems using controllers that act on the system through the boundary conditions.

Date received: April 25, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacc-15.