How suboptimal is my stochastic network controller allowed to be? Completion certificates with application to power grids hosting AI data centers

Abstract

Power grids are beginning to host AI data centers whose demand can change abruptly and in a correlated way, and operators and planners must decide whether existing controllers can absorb the resulting transients and where new flexibility is worth installing. We cast this as a question in stochastic control: how far from optimal is an implementable controller? For controlled diffusions with affine, state-independent actuation, additive and possibly degenerate noise and quadratic control cost, any Hamilton-Jacobi-Bellman (HJB) subsolution bounds the optimal cost from below and simulation bounds the deployed cost from above, so their gap certifies the permissible suboptimality. We construct subsolutions from path-integral control by completing the control geometry: enlarging the control Gramian until it matches the physical noise makes the problem linearly solvable, and its Feynman-Kac value is an automatic lower bound whose HJB residual is exactly the energy of the fictitious control. A dual noise-deflation construction can be tighter but requires a curvature condition. The geometry yields planning rules: price control authority in proportion to local noise variance, and use shadow values to guide sparse reinforcement. For nonlinear stochastic swing dynamics of the IEEE 118-bus system after a severe load loss, a simple generator controller is certified within 2.8% of optimal under homogeneous forcing; under heterogeneous forcing the gap is 39% with uniform prices and 1.2% once the same total authority is repriced, before any hardware is added.