Optimal control of antagonistic large-scale networks via signed graphons

This paper studies optimal control of opinion dynamics on antagonistic large-scale networks through a signed-graphon formulation. The interaction topology is modeled by a symmetric signed graphon, which allows cooperative and antagonistic couplings to coexist within a unified infinite-dimensional framework. On L 2 ( [ 0,1 ] ) , we introduce a controlled signed graphon Laplacian dynamics and establish well-posedness of the corresponding state equation for admissible distributed controls. We then define a graphon-level notion of structural balance and prove a measurable gauge transformation showing that every balanced signed graphon system is unitarily equivalent to a cooperative graphon consensus system. This characterization yields consensus and bipolar-consensus regimes in the balanced case and identifies graphon frustration as the obstruction to exact signed clustering in the unbalanced case. Motivated by depolarization and polarization-shaping objectives, we formulate a finite-horizon linear-quadratic tracking problem for the signed graphon dynamics. We prove existence and uniqueness of optimal controls and derive first-order optimality conditions together with an operator Riccati characterization of the optimal feedback law. Finally, for finite signed-network approximations satisfying explicit trajectory-wise residual assumptions, we prove a fixed-control transfer theorem: a graphon-synthesized control produces finite-network trajectories and costs that converge to their continuum counterparts on every fixed finite horizon. This is a consistency result for a prescribed transferred control; it does not assert that the transferred graphon controller is asymptotically optimal for the finite-network optimal control problems. The results provide a rigorous continuum control framework for antagonistic network dynamics while clearly separating finite-horizon transfer consistency from finite-network asymptotic optimality.