Math Conjectures

Generalized Belgian Chocolate Problem

OPENMajorExact constant problemProposed 1994 · Standard version

Canonical statement

Call a nonzero real polynomial Hurwitz stable if all of its complex zeros have strictly negative real part. Define
δ∗=sup⁡{δ∈(0,1):there exist nonzero Hurwitz-stable x,y∈R[s] such that(s2−2δs+1)x(s)+(s2−1)y(s) is Hurwitz stable}. \delta_* = \sup\left\{\delta\in(0,1):\begin{array}{l}\text{there exist nonzero Hurwitz-stable }x,y\in\mathbb R[s]\text{ such that}\\(s^2-2\delta s+1)x(s)+(s^2-1)y(s)\text{ is Hurwitz stable}\end{array}\right\}.
Determine δ∗\delta_* exactly.
View source LaTeX
Call a nonzero real polynomial Hurwitz stable if all of its complex zeros have strictly negative real part. Define
\[
\delta_* = \sup\left\{\delta\in(0,1):\begin{array}{l}\text{there exist nonzero Hurwitz-stable }x,y\in\mathbb R[s]\text{ such that}\\(s^2-2\delta s+1)x(s)+(s^2-1)y(s)\text{ is Hurwitz stable}\end{array}\right\}.
\]
Determine \(\delta_*\) exactly.
CTRL-001:[Blondel1994Simultaneous][WangWangYu2016Open][CharlesBoston2017Chocolate]
Control theoryMathematical optimizationAnalysis and functional analysis

Notes

The Belgian chocolate problem is a simultaneous-stabilization question posed by Blondel in 1994 [Blondel1994Simultaneous]. Call a real polynomial Hurwitz stable if all of its zeros have strictly negative real part, and for δ∈(0,1)\delta\in(0,1) ask whether there exist nonzero Hurwitz-stable polynomials x,yx,y such that (s2−2δs+1)x(s)+(s2−1)y(s)(s^2-2\delta s+1)x(s)+(s^2-1)y(s) is again Hurwitz stable. The generalized problem is to determine exactly the supremum δ∗\delta_* of the feasible values of δ\delta; Blondel's original prize instance — the prize being Belgian chocolate — was the single value δ=0.9\delta=0.9.

The instance δ=0.9\delta=0.9 was eventually solved positively, and constructive methods have pushed well beyond it: by exploiting algebraic structure in the associated global optimization, feasible values of at least 0.98083480.9808348 are known [CharlesBoston2017Chocolate]. Exact bounds are known in several fixed-degree subclasses, and the surrounding family of open simultaneous-stabilization questions is surveyed in [WangWangYu2016Open].

The unrestricted threshold, however, remains unknown: δ∗\delta_* lies somewhere between 0.98083480.9808348 and 11, and determining it exactly — the problem as stated here — is open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.