Dear Colleagues,

We continue our COLLOQUIUM.

Boris P Belinskiy, UTC

and

Sergei A Avdonin, University of Alaska Fairbanks

Tuesday, January 21, Metro 161, 2:00 pm.

Some New Developments in the Exact Control Theory

Abstract. We study the exact controllability problem for a flexible
elastic string fixed at the end points under an axial stretching
tension. The tension is a sum of two terms, a constant tension and a
variable load. We say that the string is controllable if it is possible
to find a transverse load $g(x)f(t)$ such that the string goes to the
rest. We apply the method of moments to prove our results. This method
has been widely used in control theory of distributed parameter systems
since the classical papers of H.O. Fattorini and D.L. Russell in the
late 60s to early 70s. We reduce the problem to a moment problem for the
control $f(t).$ The proof of controllability is based on an auxiliary
basis property result that is of independent interest. The main
difference between our problem of control and the previous problems is
that the coefficient of the wave equation (tension) is a function of
time. As a result, the functions that substitute non-harmonic
exponential functions even may not be found explicitly. This fact
sufficiently complicates the analysis of controllability. To our best
knowledge it is the first attempt to apply the method of moments to
equations with time dependent coefficients.

Students who have already taken the first courses in ODE and Linear
Algebra are welcome to participate.