On the Belgian chocolate problem and output feedback stabilization: Efficacy of algebraic methods

The Belgian chocolate problem asks for which values of a process parameter δ there exist three stable polynomials satisfying a certain controller design equation. For small n, earlier papers have employed global optimization techniques to obtain solutions involving polynomials of degree ≤ n, with each successive paper having a slightly larger δ. We note that these solutions converge to a critical case and we introduce algebraic methods to identify that case. The previously obtained solutions are simply approximations to this critical case, with the particular δ artificially constrained by the precision to which the authors were working. As a demonstration of the efficacy of algebraic methods, we see that the record value of δ can be increased from 0.973974 to 0.976461. We discover a surprising connection with the abc theorem for polynomials and present ideas for systematically increasing this record.