SOME SYSTEMS THEOREMS ARISING FROM THE BIEBERBACH CONJECTURE

This paper describes 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of Vasyunin and Nikol’skii which in turn is heavily dependent on the original proof by de Branges and Theorem 94 of the unpublished manuscript (Reference 10). Also one is referred to a system theoretic approach in Reference 8. One of the contributions of this paper is to identify key constraints and estimates in Reference 1 as very natural engineering systems constraints. We were extremely surprised by the extent to which this was possible after slightly modifying the class of systems treated by Vasyunin and Nikol’skii. Another contribution is to extend the generality in Reference 1 from systems with no output and with invertible input operators to conventional [A, B, CJ 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 Reference 1. While algebraically our results are more general than Reference 1, we do not have time-varying input and output spaces or unbounded operators, nor do we worry about technical issues in Hilbert space. The proof of Reference 1 may be thought of in four parts, corresponding to the four sections 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 Reference 1) is a BRL for a convex family SYSK of systems; indeed it is a robustness result of currently fashionable type. The third part develops a test to determine whether there is a uniform bound on the input-output operator in SYSK. To actually put teeth in the general systems theorem requires a strong assumption. In this case it is roughly that the extreme points of the convex set of systems are systems all of which have the same frequency response function. Under a somewhat stronger assumption the technique Vasyunin and Nikol’skii’ call chronological averaging applies to reduce the uniform bound computation for all of SYSK to solving a Riccati equation associated with just one system. The last part pertains only to ‘Lowner systems’ and, although very specialized, gives an impressive