Well-posedness of the controlled Dyson equation for general controls

Establish well-posedness of the controlled Dyson equation (partial_s m+partial_x(m(H[m]+alpha))=0) for general measurable controls alpha, beyond the currently treated cases with smooth vector fields.

Background

The paper studies the controlled Dyson equation, in which the Hilbert-transform drift is augmented by a control field alpha. Existence and uniqueness of trajectories for this equation are needed to formulate and analyze the associated optimal-control problem and its infinite-dimensional Hamilton–Jacobi–Bellman equation.

The authors explain that existing well-posedness results primarily cover smooth vector fields, whereas the optimal-control framework permits substantially more general measurable controls. Thus, determining whether the controlled equation is well posed in this broader class is explicitly identified as an unresolved issue, although the paper develops new existence results for relevant classes of controls.