Equality of the weighted relaxation and Brownian control value

Determine whether the supremum over all admissible weights of the weighted integrated-L² relaxation values V_0(η) equals the uniform-in-time Brownian control value E_*(0), that is, establish whether sup_η V_0(η)=E_*(0).

Background

The paper defines E_(κ) as the minimum essential-in-time second-moment budget of a progressively measurable Brownian drift u for which B_1+∫0^1 u_t dt≥κ almost surely. To obtain a lower bound on E(κ), it introduces the weighted relaxation V_κ(η), which minimizes the weighted integrated cost E∫0^1 η(t)u_t^2dt over the same feasible controls, where η is a positive weight integrating to one. Every such relaxation satisfies Vκ(η)≤E_*(κ), so optimizing over weights can potentially sharpen the lower bound.

At zero margin, numerical optimization of this relaxation suggests a value near 1.7373, close to but below the certified lower bound for E_*(0). The unresolved issue is whether optimizing over all admissible weights recovers the original uniform-in-time control value exactly, rather than merely providing a lower bound.