Posted Dec 1, 1995
This paper describes how one proof of the Bieberbach conjecture is remarkably parallel to considerations in robust performance of a class of systems.
The purpose of this paper is to describe the system theoretic component of the proof of the Bieberbach conjecture. We were quite surprised to find strong connections to modern robust control theory. Much of the mathematical content of this note comes directly from the paper of Vasyunin and Nikolskii [VN] which in turn is heavily dependent on the original proof by de Branges [dB] and and Theorem 94 of the unpublished manuscript [dB4]. Also the reader is refered to a system theoretic approach [dB2]. One of the contributions of this paper is to identify key constraints and estimates in [VN] as very natural engineering systems constraints. We were extremely surprised by the extent this was possible after slightly modifying the class of systems treated by [VN]. Another contribution is to extend the generality in [VN] from systems with no output and with invertible input operators to conventional [A,B,C] systems. Our objective in the paper is not to actually give a full proof of the Bieberbach conjecture but to extract the systems ideas which might be of potential use to system theorists and mathematicians. Since our goal is to make a paper easily readable to system theorists, we operate at a different level of generality than [VN]. While algebraically our results are more general than [VN], we do not have time varying input and output spaces, unbounded operators, or worry with technical issues in Hilbert space.
The [VN] proof might be thought of in four parts, corresponding to the four parts of this paper. Conceptually, the first is general systems theory and actually contains a refinement of the classical Bounded Real Lemma, BRL, which is new even in finite dimensions. The second part (after a modification of [VN]) is a BRL for a convex family SYS^K of systems; indeed it is a robustness result of currently fashionable type. The third is to develop a test to determine if there is a uniform bound on the input output operator in SYS^K. To actually put teeth in the general systems theorem requires a strong assumption. In this case it is roughly: the extreme points of the convex set of systems are systems all of which have the same frequency response function. Under a bit stronger assumption the technique [VN] call chronological averaging applies to reduce the uniform bound computation for all of SYS^K to solving a Riccati equation associated with just one system. The last part pertains only to "owner systems" and while very specialized gives an impressive example of how these general theorems apply. Here we refer the reader mostly to [VN] for an excellent rigorous proof, but sketch a formal proof and give enough details to verify how the modifications we made in parts II and III (in order to obtain conventional systems theorems) blends quickly into the [VN] line of proof.