Final Statement of the Opponent on the Dissertation
Infinite-Dimensional Linear systems, Optimal Control
and Algebraic Riccati Equations
by Kalle Mikkola
The dissertation consists of several interrelated parts.
In Part II, the theory of so called well-posed linear systems
is developed, following the recent works by G. Weiss, R. Curtain
and O. Staffans. Stability (in particular, relations between
the external and internal one), realization theory, and dynamic
[partial] output feedback are studied. Part III contains optimal
control theory, including quadratic minimization problem. On top
of that, relations between the optimization problems and solvability
of [continuous-time, integral] algebraic Riccati equations are
investigated. The high point of this part [and probably the
dissertation as a whole) is the solution of H-infinity four block
problem (in terms of two Riccati equations). A (relatively) short
Part IV is devoted to discrete time analogues of the results of
Parts II-III.
The tools used in parts II-IV include operator theory (in particular,
time invariant operators, harmonic analysis, boundary behavior of
operator valued functions from Hardy classes, corona type results,
and factorization techniques. These results are collected in Part I,
and (more than) necessary systematic background for them is given in
the appendices.
Most of the results presented in the thesis are new, at least in their
full generality. Many of them generalize the prviously known statements
about (classical) systems with rational transfer functions and
finite dimensional state spaces, but then the generalizations are never
trivial and sometimes exceptionally hard. Moreover, some results are new
even in the above mentioned classical settings.
The dissertation is written with a great attention to detail, though it
is by no means an easy reading. Two things contribute to the latter:
1) the exceptional size of the thesis and 2) the abundance of references
to the forthcoming material, that is, a high non-linearity of the exposition.
Nevertheless, even in its present form the thesis, along with its high
scientific value, is also valuable as a reference source for the control
theory community. To make his results more accessible, the author would
need to give some thought to publishing the most important parts of the
thesis in a form of several (reasonably sized) articles or maybe even
a monograph. On option would be to restrict attention to the discrete time
setting, in which the results are simpler, more complete and require
substantially less preparatory work.
During the derence, Mr. Mikkola was well prepared and answered all the
questions that were raised. His thesis is exceptional not by length only
(which was already noted) but also by the proportional, and thus very
unsual, quantity of the results obtained. With their quality being high
as well, it is my firm opinion that Mikkola's thesis is of a very
high level. I recommend that this thesis be accepted with distinction.
Espoo, October 18, 2002
Professor Ilya Spitkovsky
College of William and Mary
Department of Mathematics
P.O. Box 8795
Williamsburg, VA 23187-8795
ilya@math.wm.edu