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Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control

Qi Wei, Jianfeng Tao, Haoyang Tan, Hongyu Nie

Abstract

Numerical optimal control is commonly divided between globally structured but dimensionally intractable Hamilton-Jacobi-Bellman (HJB) methods and scalable but local trajectory optimization. We introduce the Featurized Occupation Measure (FOM), a finite-dimensional primal-dual interface for the occupation-measure formulation that unifies trajectory search and global HJB-type certification. FOM is broad yet numerically tractable, covering both explicit weak-form schemes and implicit simulator- or rollout-based sampling methods. Within this framework, approximate HJB subsolutions serve as intrinsic numerical certificates to directly evaluate and guide the primal search. We prove asymptotic consistency with the exact infinite-dimensional occupation-measure problem, and show that for block-organized feasible certificates, finite-dimensional approximation preserves certified lower bounds with blockwise error and complexity control. We also establish persistence of these lower bounds under time shifts and bounded model perturbations. Consequently, these structural properties render global certificates into flexible, reusable computational objects, establishing a systematic basis for certificate-guided optimization in nonlinear control.

Featurized Occupation Measures for Structured Global Search in Numerical Optimal Control

Abstract

Numerical optimal control is commonly divided between globally structured but dimensionally intractable Hamilton-Jacobi-Bellman (HJB) methods and scalable but local trajectory optimization. We introduce the Featurized Occupation Measure (FOM), a finite-dimensional primal-dual interface for the occupation-measure formulation that unifies trajectory search and global HJB-type certification. FOM is broad yet numerically tractable, covering both explicit weak-form schemes and implicit simulator- or rollout-based sampling methods. Within this framework, approximate HJB subsolutions serve as intrinsic numerical certificates to directly evaluate and guide the primal search. We prove asymptotic consistency with the exact infinite-dimensional occupation-measure problem, and show that for block-organized feasible certificates, finite-dimensional approximation preserves certified lower bounds with blockwise error and complexity control. We also establish persistence of these lower bounds under time shifts and bounded model perturbations. Consequently, these structural properties render global certificates into flexible, reusable computational objects, establishing a systematic basis for certificate-guided optimization in nonlinear control.
Paper Structure (15 sections, 11 theorems, 199 equations, 1 figure)

This paper contains 15 sections, 11 theorems, 199 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From Bolza optimal control to occupation measures
  4. Primal LP, dual HJB inequality, and global certificates
  5. Approximate dual feasibility and relaxed certificate bounds
  6. Featurized Occupation Measures
  7. General FOM components
  8. Certificate evaluation and residual-aware bounds
  9. Explicit FOM and its asymptotic consistency
  10. Implicit FOM and its asymptotic consistency
  11. Computational consequences of FOM
  12. Structural complexity over state-space complexity
  13. Certificates are reusable online objects
  14. Global certificate as an interface across FOM realizations
  15. Conclusion

Key Result

Proposition 1

Assume the deterministic fixed-horizon setting, so that every exact OM-feasible pair $(\mu,\mu_T)$ satisfies Let $v$ be $(\varepsilon,\varepsilon_T)$-feasible and define Then

Figures (1)

  • Figure 1: Certificate-guided replanning under a structural obstacle change.

Theorems & Definitions (33)

  • Remark 1: From transport probe to certificate
  • Proposition 1: Approximate certificates provide OM lower bounds
  • proof
  • Remark 2: Bounded dual-infeasibility preserves the HJB lower-bound role
  • Definition 1: FOM framework
  • Proposition 2: Certificate evaluation identity and realization-relative gap
  • proof
  • Remark 3: Admissibility versus local update geometry
  • Example 1: A deterministic optimal-control realization of explicit FOM
  • Remark 4: Moment--SOS as a dual-feasible explicit FOM
  • ...and 23 more