Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control

Abstract

Numerical optimal control has long been split between globally structured but dimensionally intractable Hamilton--Jacobi--Bellman (HJB) methods and scalable but local trajectory optimization. We introduce Featurized Occupation Measures (FOM), a finite-dimensional primal--dual interface for coupling numerical optimal control solvers with explicit HJB subsolutions: the certificate guides the primal search, while primal residuals tighten the certificate in a primal-dual language. Two realizations are developed. The explicit realization uses finite weak-form Liouville tests, and the implicit realization couples rollout-based search with sampled primal--dual residuals. Both are proved asymptotically consistent with the exact occupation-measure linear program under refinement, separating primal expressiveness from dual accuracy in the limit. The framework also gives structural conditions under which HJB-type certificates avoid full state-space representation. For factor graphs induced by compatible passivity-based interconnections, blockwise HJB inequalities assemble into globally feasible OM-dual certificates, and the decomposition is preserved under blockwise approximation. The curse of dimensionality is then shifted from state space to interconnection topology. Approximate certificates remain reusable under time shifts and bounded model perturbations, with explicit degradation bounds. On a static obstacle-avoidance benchmark, certificates of increasing tightness guide a sample-based optimizer toward global optima, confirming that even a coarse certificate carries useful global information.