Occupation Measure Approach

• The occupation measure approach is a framework that reformulates nonlinear control and analysis problems as infinite-dimensional convex optimization over measures.
• It employs measure-LP formulations and local occupation measures with moment-SOS hierarchies, enabling pathwise constraints and scalable semidefinite relaxations.
• The methodology provides theoretical guarantees, including no-gap results under convexity, and finds applications in stochastic, deterministic, and hybrid control settings.

The occupation measure approach is a mathematical framework that reformulates classes of nonlinear analysis and control problems—particularly in stochastic optimal control, calculus of variations, and systems analysis—as infinite-dimensional convex optimization (typically linear programming) over measures on trajectory, state, or time–space–action domains. This methodology enables rigorous encoding of pathwise constraints and dynamics, provides powerful numerical relaxation techniques via moments and sums-of-squares, and achieves global optimality or certified bounds under broad convexity conditions. It unifies numerous themes in stochastic analysis, optimization, nonlinear PDE/ODE control, and statistical inference.

1. Fundamental Definitions and Measure-LP Formulation

Occupation measures encode statistical or empirical properties of trajectories of controlled processes or functionals of random paths. Consider a stochastic controlled diffusion: $$dx_t = f(x_t, u_t)\,dt + g(x_t, u_t)\,dB_t, \quad x_0 \sim \nu_0, \ (x_t, u_t) \in X \times U.$$ The state-action occupation measure $\xi$ on $[0,T] \times X \times U$ is defined by

$$\xi(A) = \mathbb{E}_{\nu_0} \left[ \int_0^T 1_A(t, x_t, u_t) \, dt \right].$$

The terminal occupation measure $\nu$ on $X$ is given by

$\nu(B) = \mathbb{P}\{ x_T \in B \}.$

These measures satisfy the weak Kolmogorov (Liouville/Fokker–Planck) equation for all $w \in C^{1,2}$: $$\int_X w(T, x)\,d\nu(x) - \int_X w(0, x)\,d\nu_0(x) - \int_{[0,T]\times X\times U} (A w)(t, x, u)\,d\xi(t,x,u) = 0,$$ where $A$ is the extended generator: $\xi$0 This "weak" balance law—often recast in operator form—enforces the pathwise constraints at the measure level.

For deterministic or hybrid systems, occupation measures are constructed analogously by integrating indicators or observables along solutions, with dedicated measures for intermediate switching or boundary events (Rocca et al., 2017).

The nonlinear optimal control problem

$\xi$1

is recast as the infinite-dimensional linear program

$\xi$2

Related measure-LP formulations are valid for deterministic variational problems, discrete-time controlled processes, hybrid automata, and mean-field population control (Holtorf et al., 2022, Rocca et al., 2017, Han et al., 2018, Yu et al., 17 Mar 2026).

2. Local Occupation Measures and Sparse Moment-SOS Hierarchy

A recent refinement is the local occupation measure (LOM) framework (Holtorf et al., 2022), which partitions the space-time domain into cells and introduces separate occupation and terminal measures $\xi$3 for each block $\xi$4. This localizes the Liouville constraint to each block and couples local measures via transfer (flux) measures for spatial/temporal continuity.

Numerically, replace measures by their truncated moment sequences: $\xi$5 These moments must satisfy

• positive semidefiniteness of local (moment, localizing) matrices,
• blockwise Liouville constraints in moments,
• coupling constraints for continuity across cell boundaries.

The resulting moment-SOS hierarchy yields block-sparse semidefinite programming (SDP) relaxations of the original problem. The size of the system scales linearly in the number of space-time blocks, which is advantageous compared to the global approach where complexity is combinatorial in the moment degree (Holtorf et al., 2022). Tightening the grid—rather than increasing the relaxation order—achieves better accuracy with feasible computational cost.

3. Theoretical Guarantees: Convexity, No-Gap Theorems, and Gaps

Convexity: The occupation-measure LP is convex by construction. For variational and control problems, under the hypothesis that all data (cost, pointwise and integral constraints) are convex (or affine) in the relevant variables, the occupation measure relaxation is tight—no relaxation gap is present (Henrion et al., 2023, Korda et al., 2022). Specifically, under convexity,

$\xi$6

where $\xi$7 is the value of the occupation-measure LP and $\xi$8 is the value of the original problem.

No-gap characterizations extend to certain scalar or 1D problems even without full convexity, by virtue of the structure of quasiconvexity in those cases (2207.13570, Korda et al., 2022).

Relaxation gaps: In multidimensional, vector-valued, or nonconvex settings, occupation measures may produce lower bounds strictly less than the actual minimum. Specifically, the relaxation ignores certain nonlinear, nonlocal (Jensen-type) constraints (e.g., quasiconvexity or Young-measure constraints), which are required for capturing microstructure or phase-mixing (2207.13570, Korda et al., 2022). However, in nonconvex cases the occupation-measure minimum never outperforms the convex envelope minimum.

Gaps can also be introduced by integral equality constraints, even in low-dimensional settings. The occupation measure relaxation always provides meaningful certified lower bounds or captures meaningful generalized solutions (e.g., in singular or branching regimes) (Korda et al., 2022).

4. Applications: Stochastic, Deterministic, and Hybrid Control

Stochastic optimal control: The occupation-measure approach yields convex measure-LP representations for stochastic diffusions, jump processes, and mixed systems. The LOM framework gives block-sparse SDP relaxations, offering certified lower bounds and outperforms global relaxations in high-dimensional or structured state spaces (Holtorf et al., 2022). The methodology also generalizes to mean-field control of large populations, where the measure-LP encodes the empirical distribution of agent trajectories, and Frank–Wolfe type first-order methods can be employed for high scalability (Yu et al., 17 Mar 2026).

Deterministic/discrete-time systems: For polynomial discrete-time systems, the occupation-measure Liouville equation balances initial, occupation, and exit measures, reducing controller synthesis and set approximation to a convex SDP in moments, with explicit controller extraction (Han et al., 2018).

Hybrid automata: For hybrid systems, local occupation measures are defined per mode and across transitions, with Liouville constraints encoding jump and switching events. The moment-SOS hierarchy is adapted for each domain/mode and transition (Rocca et al., 2017).

Statistical estimation: Occupation measures can be empirically estimated for stochastic processes, with L²-optimality, stable asymptotics, and explicit rates of convergence in high-frequency settings (Li et al., 2013, Amorino et al., 2022).

5. Numerical Realization and Algorithmic Techniques

All variational and control formulations produce infinite-dimensional LPs or convex programs over measures. Practical implementation proceeds by:

• Truncating the measure to a finite sequence of moments (or local moments in the LOM framework),
• Writing constraints as linear matrix inequalities (moments and localizing matrices) in the SDP,
• Solving the hierarchy of SDPs (moment-SOS hierarchy) via off-the-shelf solvers (e.g., MOSEK, SDPA, SeDuMi).

Frank–Wolfe-type methods have been developed for infinite-dimensional measure optimization (OM-MFC), where each step entails solving an optimal control problem for an extremal trajectory, with guaranteed $\xi$9 convergence in the convex case (Yu et al., 17 Mar 2026).

In all cases, the convex relaxation produces lower bounds on the true value (sharp under convexity). Certified bounds are possible, provided the moment matrices satisfy positivity and localization constraints.

6. Limitations and Extensions

• The occupation-measure LP is generically tight only under strong convexity or scalar structure. In higher dimensions or for general nonconvex problems, strict relaxation gaps may occur due to omitted nonlocal nonlinear constraints, especially those arising from the theory of Young measures and quasiconvexity (Henrion et al., 2023, 2207.13570).
• The moment-SOS approach faces combinatorial growth in the size of the moment matrices with problem dimensionality and relaxation order; exploitation of structure, sparsity, and local partitioning is critical for scalability (Holtorf et al., 2022).
• For certain classes of physical problems (magnetism, elasticity, microstructure formation), the occupation-measure relaxation aligns with the convex envelope, thus providing a valuable computational tool for energy minimization and generalized solutions (Henrion et al., 2023).
• For empirical estimation, the approach permits feasible, robust nonparametric inference under minimal assumptions with high-frequency data (Li et al., 2013, Amorino et al., 2022).

7. Summary Table: Core Aspects of the Occupation Measure Approach

Aspect	Description/Implementation	Key Source(s)
Theoretical formulation	Infinite-dimensional linear program (LP) over measures subject to Liouville constraints	(Holtorf et al., 2022, Henrion et al., 2023)
Local occupation measures	Space–time partitioning, block-sparse measure and moment SDP	(Holtorf et al., 2022)
Tightness/no-gap results	Sharp under convexity or scalar structure, may have gap in nonconvex/multidim.	(Henrion et al., 2023, Korda et al., 2022)
Numerical realization	Moment-SOS hierarchy, block-sparse or local SDP, Frank–Wolfe on measure space	(Holtorf et al., 2022, Yu et al., 17 Mar 2026)
Applications	Stochastic/deterministic optimal control, hybrid systems, mean-field, variational calculus, statistical inference	(Rocca et al., 2017, Han et al., 2018, Yu et al., 17 Mar 2026, Li et al., 2013, Amorino et al., 2022)
Limitation/controversies	Gaps in nonconvex/vector problems, scalability with state/control dimensions	(2207.13570, Korda et al., 2022)

This approach provides a unified, variationally optimal, and computationally tractable pathway from complex nonlinear trajectory problems to a convex optimization framework that is robust, extensible, and enjoys strong theoretical and numerical guarantees, particularly in convex and structured systems.