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