Stochastic Optimal Control

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Latest papers 91

12 papers in the last four weeks, up 33% on the four weeks before. 0.1% of all new papers.

Jul 24, 2026

Trajectory-Regularized Stochastic Optimal Control via KL Divergence

We introduce trajectory-regularized stochastic optimal control (TRSOC), which augments standard stochastic optimal control (SOC) with a Kullback--Leibler (KL) divergence between controlled and reference trajectory distributions. Using Girsanov's theorem, the trajectory KL reduces to a quadratic drift mismatch penalty, yielding a modified running cost that preserves the dynamic programming (DP) structure. We derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation and characterize the optimal policy. In the linear-quadratic (LQ) setting, the formulation admits a closed-form solution with an augmented control cost. Experiments show that the regularization parameter induces a trade-off between performance-driven and reference-preserving behavior, including cases with reference dynamics learned from offline data.

Mintae Kim, Koushil Sreenath

Hybrid Robotics, UC Berkeley, CA 94720, USA

Stochastic Optimal Control

Jul 20, 2026

Generalised Bellman recurrence and three dualities in sequential decision-making

What gives the Bellman equation its form? We show that the recursive properties of optimal value functions follow from three conditions: that the dynamics decomposes through sufficient statistics, that the return decomposes recursively, and that the aggregation of uncertainty is compatible with both. When all three conditions hold on a common state, the Bellman equation arises from their mutual consistency; when one fails, tractability can often be recovered by augmenting the state or by deforming return or dynamics. The same conditions are shown to give rise to three dualities: one between probability and return, one between return and aggregation, and one between aggregation and probability. Our framework reveals these dualities as arising from a single construction, unifying methods developed separately across reinforcement learning, control, and decision theory.

Fernando E. Rosas, David Hyland, Daniel Polani

Department of Informatics, University of Sussex · Department of Computer Science, University of Oxford · Department of Computer Science, University of Hertfordshire

Markov Decision Processes Sequential Decision Making Stochastic Optimal Control Dynamic Programming

Jul 18, 2026

Relative Entropy-Bounded Ambiguous Chance Constraints for Robust Planning in Nonlinear Systems

We consider defining risk probability in stochastic control problems under distribution ambiguity. Current approaches for chance-constrained control typically assume that the true state distribution is known and Gaussian distributed. These assumptions are not amenable to many real-world engineering applications where system dynamics are nonlinear and only approximately modeled. In this work, we define a distribution ambiguity set and, with a variational expression for exponential integrals, bound the expected risk value under an unknown distribution that resides within a relative entropy distance of a nominal Gaussian reference distribution. Our bound recovers the reference risk value in the zero-divergence limit. A method is presented to determine the relative entropy distance defining the ambiguity set that is a function of the reference covariance evolution and second-order dynamical truncation errors. The resulting contributions provide a framework for handling distributional ambiguity in nonlinear covariance steering problems. A stochastic spacecraft guidance example is presented to demonstrate our contributions.

Trevor N. Wolf, Jay W. McMahon

Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO 80303, USA

Chance-Constrained Optimization Stochastic Optimal Control Distributionally Robust Optimization

Jul 13, 2026

LQG solution for POMDP without estimating states: A minimum variance approach

This paper investigates the control of discrete-time linear time-invariant (LTI) systems subject to incomplete and corrupted measurements. Specifically, we focus on designing a Linear Quadratic Gaussian (LQG) controller without relying on explicit state estimation. By leveraging minimum variance duality, our approach allows the current control input to be represented as a linear function of available measurements and previously applied inputs, successfully reducing the task to a tractable deterministic optimization problem. We provide theoretical justification for this framework and demonstrate its practical effectiveness through numerical experiments.

Ranjan Sarkar, Prabhat K. Mishra

Department of Artificial Intelligence at Indian Institute of Technology Kharagpur, West Bengal, India

Stochastic Optimal Control Feedback Control Partially Observable Markov Decision Processes

Jul 5, 2026

Deep Learning for Dynamic Programming with Recursive Utility

We propose the first deep learning algorithm, the Certainty Equivalent Learning (CEL) algorithm, for solving high-dimensional discrete-time dynamic programming problems with recursive utility. Dynamic programming with recursive utility is numerically challenging because the recursive utility does not have an explicit representation and the Bellman equation contains a certainty equivalent that is difficult to evaluate. The CEL algorithm learns this certainty-equivalent value directly with neural networks and jointly approximates value functions, policy functions, and certainty-equivalent functions. The CEL algorithm is mesh-free and simulation-based, allowing high-dimensional state and control spaces, and does not rely on Euler equations, first-order conditions, or differentiability of the state transition function. The CEL algorithm also works for dynamic programming problems with expected utility as expected utility is a special case of recursive utility. We apply the CEL to discounted linear exponential quadratic Gaussian control, small-noise robust control, Epstein-Zin DSGE, and multivariate strategic asset allocation problems. Compared with closed-form and VFI-based benchmarks, the CEL delivers accurate value and policy approximations, remains effective in high-dimensional problems, achieves accuracy comparable to VFI in the small-noise robust-control case, and produces out-of-sample Bellman errors and Euler or first-order residuals that are in the range from 1.0e-4 to 1.0e-3 for most problems.

Xianhua Peng, Wu Guo

HSBC Business School, Peking University, University Town, Nanshan District, Shenzhen, 518055, China.

Stochastic Optimal Control Dynamic Programming

Jul 4, 2026

Finite-Sample Closed-Loop Stability of Model Predictive Path Integral Control for Linear Time-Invariant Systems

We establish finite-sample closed-loop stability guarantees for Model Predictive Path Integral (MPPI) control applied to discrete-time Linear Time-Invariant (LTI) systems with additive Gaussian process disturbances. The key observation is that, for unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law has the same first control action as the infinite-horizon LQR law for every planning horizon. Thus, finite-sample MPPI can be analyzed as a stochastic perturbation of LQR. First, we show that the MPPI control law approximates the LQR feedback with high probability. The approximation error decomposes into a Monte Carlo term that decreases with the sample count and an infinite-sample temperature bias that persists at finite temperature but vanishes as the temperature is reduced. The resulting constants are written in terms of the horizon-dependent stacked cost matrices, making explicit that the finite-sample certificate is parametrized by the selected planning horizon. Second, we use a Lyapunov perturbation argument to prove practical exponential stability in expectation. On sample paths that remain in a compact Lyapunov sublevel set over a finite operating horizon, the expected state norm decays exponentially up to three residual floors: a process-noise floor, an MPPI approximation floor, and a confidence floor from the per-step sampling failure probability. The sufficient sample threshold is explicit and computable from the DARE solution, LQR stability margin, MPPI sampling parameters, temperature, and planning horizon. In the joint limit of infinite samples and vanishing temperature bias, the result recovers the stochastic LQR stability bound.

Hyung-Jin Yoon, Hunmin Kim

Department of Mechanical and Nuclear Engineering, Tennessee Technological University, Cookeville, TN, USA

Dynamical Systems Stochastic Linear Dynamical Systems Lyapunov Stability Stochastic Optimal Control MPPI Control

Jun 27, 2026

Stochastic Optimal Control Sampling for Diffusion Inverse Problems

Benefiting from the strong ability to capture data distributions, diffusion models have become powerful tools for solving image inverse problems. The key is to controllably steer the sampling trajectory toward the measurements while respecting the diffusion prior. In this work, we introduce Stochastic Optimal Control Sampling (SOCS), which models the denoising process as a dynamical system and injects control signals via SOC. Previous SOC-based approach addresses inverse problems by optimizing over the entire trajectory, which is computationally expensive. In contrast, we derive a closed-form control update and apply it at each sampling step, pulling the measurement-consistent clean prediction back onto the denoising flow. In SOCS, we can readily modulate the control strength to align with the diffusion model's native capabilities and thereby enhance perceptual quality. Our method is compatible with a variety of linear stochastic differential equation backbones. Extensive experiments across a broad spectrum of image inverse tasks demonstrate that SOCS achieves accurate measurement-aligned reconstructions with improved visual fidelity and stronger quantitative performance.

Jie Zhang, Youmei Qiu, Hanling Tian +3

School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai, China · Shanghai Key Laboratory of Flexible Medical Robotics, Tongren Hospital, Institute of Medical Robotics, Shanghai Jiao Tong University, Shanghai, China

Stochastic Optimal Control Diffusion-Based Inverse Problems Diffusion Posterior Sampling Image Inverse Problems

Jun 25, 2026

Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data

This paper addresses model-free continuous-time mean-field control in a setting where the population dynamics evolve continuously according to an unknown McKean-Vlasov stochastic differential equation, while only discrete-time transition data are available. In the model-based formulation, policy evaluation is naturally described by a stationary Hamilton-Jacobi-Bellman equation on

\mathcal P_2(\mathbb R^d)

, but this equation involves the drift and diffusion coefficients of the controlled McKean-Vlasov dynamics, which are not identifiable when only discrete-time data are available. On the other hand, a direct reduction to a time-discrete Bellman equation avoids the non-identifiability issue but loses the differential equation structure. To bridge these two viewpoints, we introduce a Mean-Field-PhiBE (MF-PhiBE), which incorporates discrete-time transition information into a continuous-time PDE on the Wasserstein space. The MF-PhiBE replaces the unknown infinitesimal drift and covariance in the policy-evaluation equation by one-step estimators computed from data, while preserving the generator structure of the McKean-Vlasov HJB equation. We also derive a policy-gradient theorem for entropy-regularized randomized feedback policies, expressing the actor direction through an action-wise infinitesimal advantage and the score of the policy. Combining these two ingredients yields a model-free actor-critic method. We prove a first-order consistency estimate showing that the value induced by an optimal MF-PhiBE policy approximates the optimal continuous-time value with an error of order

Δt

. In the linear-quadratic case, we show our approximation achieves second-order accuracy with only one-step data. Numerical experiments on an LQR benchmark and a crowd-aversion problem illustrate the proposed framework.

Erhan Bayraktar, Martin Hernandez, Qinxin Yan +1

Department of Mathematics, University of Michigan, Ann Arbor, MI, USA. · Department of Statistics and Data Science, University of California, Los Angeles, CA, USA. · Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ, USA.

Reinforcement Learning Stochastic Optimal Control Continuous-Time RL Model-Free RL Actor-Critic Methods

Jun 18, 2026

QMaxCal: Path-Space Regularization for Open Quantum Control via Girsanov's Theorem

Reliable quantum control in the presence of decoherence requires policies that combat the effect of environmental noise on the controlled dynamics. Open quantum systems under continuous monitoring generate classical measurement records whose drift depends on the noise experienced by the system; the records of two evolutions sharing the same decoherence channels differ only in this drift, so Girsanov's theorem yields a closed-form, differentiable estimator of the KL divergence between their trajectory distributions. We instantiate this estimator with two physically motivated reference measures, yielding two regularizers that both drive the system toward states where the effects of decoherence are minimal: the Wiener KL (KL_W), which is empirically more effective under certain conditions on the noise model, and the drift-variance regularizer (R_DV), which works for all noise models. Both are qualitatively distinct from existing penalties on control fluence or smoothness: they penalize the observable consequences of control on the decoherence channels rather than the control amplitude itself. The regularizers outperform unregularized gradient-based and reinforcement-learning baselines across a range of open quantum systems -- including single- and multi-qubit benchmarks and a multi-qubit chain calibrated to a published snapshot of the IBM Kingston processor -- along several axes of evaluation: final-state fidelity, robustness to mismatch in the assumed noise model (gains grow from +17 pp at training noise to +27 pp under 2.5x noise mismatch), and occupation of forbidden states. The regularizers reduce infidelity by up to 50%, with ~16% gains on the calibrated IBM Kingston chain.

Merijn Moody, Zier Mensch, Miranda C. N. Cheng +2

Dutch Institute for Emergent Phenomena, University of Amsterdam, Netherlands · Institute of Physics, University of Amsterdam, Netherlands · Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands +4

Stochastic Optimal Control

Jun 13, 2026

DiRecT: Safe Diffusion-Based Planning via Receding-Horizon Denoising

Diffusion models have emerged as powerful tools for planning and control by learning multimodal distributions over actions and trajectories. Yet reliable inference-time safety enforcement remains a key barrier to their deployment in safety-critical tasks. Existing approaches typically project each denoising iterate onto the feasible set, even though constraints are defined only on the final clean trajectory. Enforcing feasibility on noisy intermediate samples can therefore overconstrain the sampling dynamics, substantially degrading sample quality. To address this limitation, we introduce DiRecT (Diffusion-based planning via Receding-horizon denoising with Terminal constraints), a training-free algorithm for constrained sampling from diffusion models via stochastic optimal control (SOC). DiRecT enforces constraints only on the final clean sample, avoiding unnecessary restrictions on the intermediate denoising dynamics. Inspired by model predictive control, we derive a principled receding-horizon surrogate for the otherwise intractable constrained SOC formulation, yielding an efficient algorithm that cleanly separates stochastic denoising from constraint satisfaction, progressively steering samples toward feasible final trajectories without distorting the learned diffusion dynamics. Furthermore, DiRecT is highly flexible: it can leverage off-the-shelf or domain-specific optimizers, incorporate priors over environment dynamics, and optimize additional soft rewards. Extensive experiments on safe planning benchmarks demonstrate that DiRecT substantially improves deployment safety and task performance over existing diffusion-based planning baselines.

Paolo Giaretta, Zeyang Li, Navid Azizan

MIT

Constrained Motion Planning Diffusion-Based Planning Stochastic Optimal Control Diffusion Sampling Safe Motion Planning

Jun 10, 2026

Deterministic Policy Gradient for Learning Equilibrium in Time-Inconsistent Control Problems

In this paper, we develop a continuous-time model-free reinforcement learning algorithm to learn deterministic equilibrium policies in general time-inconsistent control problems. Utilizing the extended Hamilton-Jacobi-Bellman system, we recast the original time-inconsistent problem into an equivalent two-stage problem. In the first stage, for given auxiliary functions, we employ the deterministic policy gradient approach to learn an optimal policy in an auxiliary time-consistent control problem. In the second stage, given the updated policy, we exploit the inner fixed point iterations and some martingale characterizations to learn the auxiliary functions. As a theoretical contribution, we provide some mild model assumptions and establish the convergence of inner fixed point iterations. By repeating this actor-critic style of iterations across two stages, our algorithm aims to learn the equilibrium under different sources of time-inconsistency in a unified manner. The superior effectiveness of the proposed algorithm are illustrated in two classical financial applications with time-inconsistency: mean-variance portfolio management and optimal tracking portfolio under non-exponential discounting.

Xin Guo, Yijie Huang, Xiang Yu

Department of Industrial Engineering and Operations Research, University of California, Berkeley, USA. · Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong.

Policy Gradient Stochastic Optimal Control Continuous-Time RL Model-Free RL Deep Deterministic Policy Gradient Policy Learning+1

May 31, 2026

Autopilot-Preserving Residual Q-Learning with HJB-Inspired Finite-Action Risk Filtering for Fixed-Wing UAV Command Supervision

A fixed-wing UAV must hold airspeed, altitude, and heading references under wind, gusts, and turbulence, channels coupled so that correcting one can degrade another. Classical autopilots stabilize the airframe well but adapt poorly when a hard crosswind meets an aggressive turn, while reinforcement-learning (RL) policies acting directly on the surfaces concentrate exploration risk at the actuator interface. We place a learned supervisor above an unchanged autopilot rather than inside it: it selects a residual from a finite, bounded action set on the commanded airspeed, altitude, and heading; the modified reference is projected into an admissible command envelope before reaching the autopilot, which stays the only actuator-facing controller. What is new is how the residual is chosen. HJB residual scores candidates with a semi-discrete value-iteration critic in the spirit of the Hamilton-Jacobi-Bellman (HJB) equation, ranks them by a no-op-relative Hamiltonian advantage, and filters them through a control-Lyapunov- and control-barrier-inspired finite-action shield that always keeps a no-op fallback. On a shared 12-state runtime holding the plant, autopilot, and actuator model fixed, so the comparison is at the package level, HJB residual lowers mean RMS path-tracking error to 44.809 m, against 338.617 m for the baseline autopilot and 88.809 m for a tabular-Q residual, an 86.77% reduction over the baseline and 49.54% over Q-learning. The gain concentrates where the baseline fails worst and comes with a measured rise in airspeed error, so no method dominates every metric. We present this autopilot-preserving residual command-supervision design and benchmark with its trade-offs reported intact.

Mehmet Iscan, Batuhan Temiz

PythaLab, Yildiz Technical University, Istanbul, Turkey · Autopilot Systems Chief Engineer, Turkish Aerospace (TUSAŞ), Ankara, Turkey

Aerial Robotics Stochastic Optimal Control Residual Policy Learning Safe RL

May 31, 2026

A Per-Component Diagnostic Protocol for Neural HJB-PIDE Solvers under Control-Dependent Lévy Jumps

We propose a five-step diagnostic protocol for residual-trained neural HJB-PIDE solvers with control-dependent Lévy jumps, targeting a general failure mode of neural PDE methods: a learned solution can match headline scalar diagnostics while miscomputing an operator inside its training loss. The protocol pairs each neural solve with at least one from-scratch independent reference, decomposes the Hamiltonian into drift, diffusion, compensator, and nonlocal-integral components across a u-grid, and compares the value function and its low-order derivatives over a (t,x) grid before any argmax comparison. Applied to a standard CRRA-Merton-Variance-Gamma benchmark, it isolates a missing 1/2-mixture factor in the neural method's importance-proposal density that scaled the nonlocal integral by exactly half - a textbook signature of a constant proposal scale error, invisible to longer training, grid refinement, and truncation sweeps. With the bug corrected, four references - two finite-difference solvers with disjoint discretizations, the neural solver, and a semi-analytic scalar baseline obtained from CRRA homogeneity - agree on the optimal control to within ~2%. The constant-coefficient CRRA benchmark collapses by homogeneity to a scalar maximization, so the scalar baseline is the efficient method here; the contribution is the protocol, applicable in principle to non-homogeneous and higher-dimensional settings where neural HJB-PIDE solvers are genuinely needed. The episode is a concrete instance of a broader neural-PDE verification failure: pointwise agreement of a learned value or control can coexist with a systematically wrong nonlocal operator, so per-component and surface-level checks are needed before trusting the argmax policy.

R. Drissi

Neural PDE Solvers Stochastic Optimal Control

May 31, 2026

Prospect-Theory Behavior from Bellman Optimality in MDPs with Catastrophic States

We study risk-neutral control in Markov decision processes with an absorbing catastrophic state. Even though rewards are linear and the agent has no utility curvature, probability weighting, or framing dependence, standard Bellman optimality produces three prospect-theory-like signatures: an S-shaped value-function profile (convex near catastrophe, concave in the far field), an endogenous loss-sensitivity coefficient

λ^*(S) > 1

, and a reflection-effect policy reversal. Across 495 configurations, the optimal policy plays safe near catastrophe in positive-drift (growth) regimes despite the risky action's higher immediate expected value, and plays risky near catastrophe in negative-drift (decline) regimes despite the safe action's lower immediate expected loss. We derive a closed-form expression for the asymptotic loss-aversion plateau

\barλ

that depends only on win probability

p

, payoff asymmetry

r = |Δ_\ell/Δ_w|

, and discount factor

β

, and matches numerical solutions to

R^2 = 0.999

. The mechanism does not require asymmetric payoffs. Across a sweep of

(p,β)

at three asymmetry levels, the asymmetry share of

\barλ

above unity has median 4.6% at

r = 1.25

and rises to 13.9% at

r = 2

, with the boundary contribution exceeding the asymmetry contribution in every cell tested. The phenomena persist under tabular Q-learning (a model-free agent reproduces

V^*

at correlation 0.98 in growth and 1.00 in decline) and under stochastic transitions with Gaussian, heavy-tailed Student-

t_3

, and asymmetric skew-normal noise up to 50% of the step size, where the asymptotic plateau tracks the closed-form prediction within 0.41% for safe-channel noise and within 9.6% for risky-channel or both-channel noise. These results identify absorbing failure states as a sufficient structural mechanism for prospect-theory-like behavior under optimal control.

Yujiao Chen

Massachusetts Institute of Technology · Cambridge, MA 02139, USA

Markov Decision Processes Stochastic Optimal Control Optimal Control Decision-Making under Uncertainty

May 26, 2026

Sample Complexity of Policy Gradient for Log-Growth Control

We study the sample complexity of policy gradient for log-growth control -- the problem of learning, from observed state transitions, a feedback gain that optimally stabilizes a scalar linear system driven through a multiplicative-noise actuation channel. The objective

J(K) = \mathbb{E}[\log|1+BK|]

is the top Lyapunov exponent of the closed loop. This problem carries a structural difficulty we call the cusp obstruction: the optimal gain

K^*

always places the noise singularity

b_{\rm sing}(K) = -1/K

in the interior of the support. At this singular optimum the policy gradient exists only as a Cauchy principal value, not as a Lebesgue integral, and the natural single-sample gradient estimator has infinite variance. Standard first-order stochastic-optimization analysis is thus inapplicable at the optimum, and merely smoothing the objective does not resolve the difficulty. The obstruction, however, has an exploitable symmetry: the Cauchy kernel is an odd function of the displacement from the moving pole, so pairing each observation with its reflection through the pole cancels the divergent part. This one cancellation simultaneously controls the population curvature, the gradient-estimator variance, and the bias incurred when the noise density is estimated. Combining these bounds with a closed-form single-transition gradient oracle, we prove that projected mini-batch policy gradient, initialized in any compact subset of the stabilizing region, attains total sample complexity

\tilde{O}(1/η)

when the noise density is known and

\tilde{O}(η^{-(2s+1)/(2s)})

when it must be estimated, for

C^s

noise densities with

s \geq 2

.

Qiuhua Pan, Yukai Shen, Liwei Zhang +2

State Key Laboratory of Submarine Geoscience, School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China · Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China · Shanghai Key Laboratory of Perception and Control in Industrial Network Systems, Shanghai 200240, China +1

Stochastic Linear Dynamical Systems Stochastic Optimal Control Policy Gradient Methods Optimal Control Sample-Efficient RL

May 24, 2026

Lifted Schrödinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions

We study stochastic density control between Gaussian-mixture endpoint distributions under Brownian prior dynamics. Since the direct Schrödinger bridge between Gaussian mixtures is generally not available in closed form, we introduce a lifted path-space construction in which each trajectory is augmented with a source--target component label. Consequently, the problem decomposes into Gaussian component-to-component Schrödinger bridges with explicit marginal, drift, and cost formulas, while the mixture-level assignment reduces to a finite-dimensional entropic coupling problem with a Sinkhorn scaling form. We then analyze the projection obtained by discarding or forgetting the label. By construction, the projected law satisfies the original Gaussian-mixture endpoint constraints, but its relative entropy generally differs from the lifted relative entropy by a nonnegative conditional label-information gap. This gap reveals a path-space obstruction: the lifted optimizer cannot, in general, be identified with the direct unlabeled Schrödinger bridge after projection. We also derive the posterior-averaged Markov drift associated with the projected marginal flow, prove a kinetic-energy upper bound, and identify a common path-potential condition under which the projection gap vanishes. Several numerical illustrations showing density and shape control are recorded for a self-contained exposition.

Siddhartha Ganguly, George Rapakoulias, Panagiotis Tsiotras

S. Ganguly, G. Rapakoulias, and P. Tsiotras are with Daniel Guggenheim School of Aerospace, Georgia Institute of Technology, ½ Atlanta, USA.

Stochastic Optimal Control Schrödinger Bridges Optimal Transport

May 21, 2026

Generative Modeling by Value-Driven Transport

We propose a new framework for generative modeling based on a discrete-time stochastic control formulation of measure transport. Adapting classic results from control theory, we formulate our problem as a linear program whose dual variables correspond to the optimal value function of the control problem, which directly encodes the optimal control policy. Exploiting this LP formulation, we develop an efficient simulation-free primal-dual algorithm for computing approximately optimal value functions and the associated value-driven transport (VDT) policies which approximate the true optimal policy. We show that well-trained VDT policies enjoy numerous favorable properties in comparison with other state-of-the-art methods based on flows, diffusions, or Schrödinger bridges: they lead to straight transport paths which can be simulated quickly and robustly, and can be enhanced in all the same ways as diffusion and flow-based models (e.g., conditional generation, classifier-free guidance, unpaired data-to-data translation are all easy to incorporate). We evaluate our methodology in a range of experiments, with results that indicate strong performance and good potential for scalability.

Pablo Moreno-Muñoz, Adrian Müller, Gergely Neu

Universitat Pompeu Fabra · Barcelona, Spain · ETH Zürich +2

Value Function Estimation Stochastic Optimal Control Generative Modeling Optimal Transport

May 20, 2026

Zero-shot adaptation to order book dynamics

We describe an adaptive market-making architecture that preserves the analytical structure of the Avellaneda--Stoikov framework while introducing a successor measure-style adaptation mechanism. In our paper we keep Avellaneda--Stoikov fast Hamilton--Jacobi--Bellman structure and make it adaptive to changing market regimes and trading objectives. The central idea is to separate market dynamics from the trading objective. The market state determines a low-dimensional set of Avellaneda--Stoikov parameters, while recent realized rewards determine a low-dimensional objective vector. The HJB forward map then converts this objective into optimal bid and ask quotes through a scalarization of future reward features.

Arip Asadulaev

Sequential Decision Making Stochastic Optimal Control Optimal Control Adaptive Control Finance Algorithmic Trading

May 20, 2026

Beyond the Bellman Recursion: A Pontryagin-Guided Framework for Non-Exponential Discounting

Most value-based and actor--critic reinforcement learning methods rely on Bellman-style recursions, yet these recursions collapse under non-exponential discounting common in human preferences and survival processes. We show the breakdown is structural: exponential discounting sits at a fragile intersection of multiplicativity and time homogeneity, and violating either property breaks standard dynamic programming. To overcome this, we propose Pontryagin-Guided Direct Policy Optimization (PG-DPO), a variational framework that abandons recursion and couples the Pontryagin Maximum Principle with Monte Carlo rollouts via an Adjoint-MC projection enforcing pointwise Hamiltonian maximization. Across multi-dimensional hyperbolic and survival-discount benchmarks, PG-DPO improves accuracy and stability where equation-driven solvers and critic-based baselines diverge.

Hojin Ko, Jeonggyu Huh

Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea

Reinforcement Learning Stochastic Optimal Control Policy Optimization Policy Learning Optimal Control

May 14, 2026

Language Generation as Optimal Control: Closed-Loop Diffusion in Latent Control Space

This work reformulates language generation as a stochastic optimal control problem, providing a unified theoretical perspective to analyze autoregressive and diffusion models and explain their limitations (Efficiency-Fidelity Paradox, Irreversibility Error Propagation, Optimization Tractability and Fidelity) in terms of combination of trajectory singularity, adjoint state vanishing, and gradient absence. To address these issues, we approximate the solution to the Hamilton-Jacobi-Bellman (HJB) equation, yielding an optimal policy that acts as a closed-loop controller. To bypass the intractability of directly solving the HJB PDE, we employ Flow Matching as the optimal trajectory solver within the rectified latent control space. This allows our Manta-LM with Global Integral Operator to approximate the global vector field, effectively realizing a model that simultaneously achieves high-fidelity text generation and efficient, low-cost parallel sampling. Empirically, our method achieves strong performance on language modeling and conditional generation tasks, while exhibiting improved stability, efficiency, and controllability.

ZiYi Dong, Yuliang Huang, Weijian Deng +3

Sun Yat-sen University, Guangzhou, Guangdong, China · Australian National University, Canberra, ACT, Australia · Tsinghua University, Beijing, China

Language Model-Based Control Stochastic Optimal Control Diffusion Sampling Diffusion Language Models Optimal Control

May 12, 2026

Efficient Adjoint Matching for Fine-tuning Diffusion Models

Reward fine-tuning has become a common approach for aligning pretrained diffusion and flow models with human preferences in text-to-image generation. Among reward-gradient-based methods, Adjoint Matching (AM) provides a principled formulation by casting reward fine-tuning as a stochastic optimal control (SOC) problem. However, AM inevitably requires a substantial computational cost: it requires (i) stochastic simulation of full generative trajectories under memoryless dynamics, resulting in a large number of function evaluations, and (ii) backward ODE simulation of the adjoint state along each sampled trajectory. In this work, we observe that both bottlenecks are closely tied to the non-trivial base drift inherited from the pretrained model. Motivated by this observation, we propose Efficient Adjoint Matching (EAM), which substantially improves training efficiency by reformulating the SOC problem with a linear base drift and a correspondingly modified terminal cost. This reformulation removes both sources of inefficiency; it enables training-time sampling with a few-step deterministic ODE solver and yields a closed-form adjoint solution that eliminates backward adjoint simulation. On standard text-to-image reward fine-tuning benchmarks, EAM converges up to 4x faster than AM and matches or surpasses it across various metrics including PickScore, ImageReward, HPSv2.1, CLIPScore and Aesthetics.

Jeongwoo Shin, Dongsoo Shin, Yuchen Zhu +5

Seoul National University · Georgia Institute of Technology · Sungkyunkwan University

Diffusion Model Alignment Stochastic Optimal Control Diffusion Model Fine-Tuning RL for Diffusion Models T2I Diffusion Models

May 12, 2026

Gradient-Free Noise Optimization for Reward Alignment in Generative Models

Existing reward alignment methods for diffusion and flow models rely on multi-step stochastic trajectories, making them difficult to extend to deterministic generators. A natural alternative is noise-space optimization, but existing approaches require backpropagation through the generator and reward pipeline, limiting applicability to differentiable settings. To address this, here we present ZeNO (Zeroth-order Noise Optimization), a gradient-free framework that formulates noise optimization as a path-integral control problem, estimable from zeroth-order reward evaluations alone. When instantiated with an Ornstein--Uhlenbeck reference process, the update connects to Langevin dynamics implicitly targeting a reward-tilted distribution. ZeNO enables effective inference-time scaling and demonstrates strong performance across diverse generators and reward functions, including a protein structure generation task where backpropagation is infeasible.

Jeongsol Kim, Hongeun Kim, Jian Wang +1

KAIST AI1 · Snap Inc.2

Diffusion Model Alignment Stochastic Optimal Control Zeroth-Order Optimization Generative Modeling Derivative-Free Optimization

May 10, 2026

Mutual Information Optimal Density Control of Linear Systems and Generalized Schrödinger Bridges with Reference Refinement

We consider a mutual information (MI) regularized optimal density control of a discrete-time linear system. MI optimal control has been utilized for exploration in reinforcement learning and privacy protection in control. MI regularization induces stochasticity in the policy, which poses challenges for applications of MI optimal control in safety-critical scenarios. To remedy this situation, we impose Gaussian density constraints at specified times to directly control state uncertainty. For this MI optimal density control problem, we propose an alternating optimization algorithm and investigate its convergence properties. In addition, we reveal a relationship between the MI optimal density control problem and a so-called generalized Schrödinger bridge problem associated with the discrete-time linear system. Based on the results of MI optimal density control and this relationship, we also investigate alternating optimization for the Schrödinger bridge problem and its convergence properties.

Shoju Enami, Kenji Kashima

Graduate School of Informatics, Kyoto University, Kyoto, Japan

Stochastic Optimal Control Schrödinger Bridges Distribution Matching Constrained Optimization Optimal Control

May 8, 2026

Monotone Neural Policy Iteration for High-Dimensional First-Order Hamilton--Jacobi--Bellman Equations

We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics. Centered differences and an artificial viscosity

Nh=O(h)

define a monotone operator evaluated through

2d+1

shifted network queries; policy iteration solves the resulting Bellman equation without a tensor grid. At fixed

h

, the sharp componentwise condition

\max_i|f_i|\le2N

turns every frozen-policy operator into a nearest-neighbor Markov-chain generator with a policy-independent total jump rate. Uniformization gives whole-space well-posedness for measurable feedbacks, an explicit Poisson-tail bound on the numerical domain of dependence, and boundary-free localization. The representation also yields a posteriori policy-evaluation bounds that account for residual and learned-model errors. A greedy-gap analysis controls inexact policy iteration at fixed

h

; a separate consistency estimate connects the semi-discrete equation to the continuous HJB equation. Experiments reproduce the extremal tail, show rates consistent with

O(\sqrt h)

and nearly

h

-independent exact-policy-iteration decay, and assess empirical estimator effectivity. A nonsmooth example shows that the continuous residual can miss a non-viscosity solution, whereas the shifted residual detects the defect. Further tests provide a structured interval-verified certificate calibration, an early-budget benefit of policy freezing for bang-bang control, and learned-dynamics diagnostics. A structured nonlinear problem with active compact-control constraints is tested against a manufactured semi-discrete reference through

d=1024

.

Minseok Kim, Yeongjong Kim, Namkyeong Cho +1

Department of Applied Artificial Intelligence, Seoul National University of Science and Technology, Seoul, Republic of Korea. · Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea. · Department of Mathematical Finance and Big Data, Gachon University, Seongnam, Republic of Korea +1

Neural PDE Solvers Stochastic Optimal Control Policy Iteration Optimal Control

May 5, 2026

Globally Solving Unbalanced Optimal Transport and Density Control for Gaussian Distributions

In this article, we study unbalanced optimal transport (UOT) and establish a control-theoretic dynamical extension, which we call the unbalanced density control (UDC), for a class of Gaussian reference measures. In the static setting, we consider UOT with quadratic transport cost and Kullback--Leibler penalties on the marginals relative to prescribed Gaussian measures. We show that the infinite-dimensional variational problem admits an exact Gaussian reduction, yielding a finite-dimensional optimization over masses, means, and covariances, together with a closed-form expression for the optimal transported mass. We then formulate UDC for discrete-time linear systems, where the initial and terminal state measures are imposed softly through KL penalties and the intermediate evolution is governed by controlled linear dynamics with quadratic control cost. For this problem, we prove that any feasible solution can be replaced, without loss of optimality, by a Gaussian initial measure and an affine-Gaussian control policy. This leads to an exact finite-dimensional reformulation and, after a standard covariance-steering lifting, to an SDP-based optimization for fixed mass, again coupled with a closed-form mass update. We further establish existence of optimal solutions and identify a sufficient condition under which the affine-Gaussian UDC policy is deterministic. These results provide globally optimal solution methods for both Gaussian UOT and Gaussian UDC. Finally, we illustrate our results with several numerical examples.

Haruto Nakashima, Siddhartha Ganguly, Kenji Kashima

The Applied Mathematics and Physics Department, Graduate School of Informatics, Kyoto University, Kyoto, Japan · Daniel Guggenheim School of Aerospace, Georgia Institute of Technology, Atlanta, USA

Semidefinite Programming Stochastic Optimal Control Optimal Control Optimal Transport

May 5, 2026

Adaptive Estimation and Optimal Control in Offline Contextual MDPs without Stationarity

Contextual MDPs are powerful tools with wide applicability in areas from biostatistics to machine learning. However, specializing them to offline datasets has been challenging due to a lack of robust, theoretically backed methods. Our work tackles this problem by introducing a new approach towards adaptive estimation and cost optimization of contextual MDPs. This estimator, to the best of our knowledge, is the first of its kind, and is endowed with strong optimality guarantees. We achieve this by overcoming the key technical challenges evolving from the endogenous properties of contextual MDPs; such as non-stationarity, or model irregularity. Our guarantees are established under complete generality by utilizing the relatively recent and powerful statistical technique of

T

-estimation (Baraud, 2011). We first provide a procedure for selecting an estimator given a sample from a contextual MDP and use it to derive oracle risk bounds under two distinct, but nevertheless meaningful, loss functions. We then consider the problem of determining the optimal control with the aid of the aforementioned density estimate and provide finite sample guarantees for the cost function.

Riddhiman Bhattacharyya, Sayak Chakrabarty, Imon Banerjee

Department of Statistics, University of California, Santa Cruz Santa Cruz, California, USA · Department of Computer Science Northwestern University Evanston, IL, USA · Department of Industrial Engineering and Management Sciences Northwestern University Evanston, IL, USA

Non-Stationary RL Markov Decision Processes Stochastic Optimal Control Offline RL Interval Markov Decision Processes Optimal Control+1

May 4, 2026

Feedback Motion Planning for Stochastic Nonlinear Systems with Signal Temporal Logic Specifications

We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.

Liqian Ma, Zishun Liu, Glen Chou +1

Georgia Institute of Technology, Atlanta, GA 30332

Constrained Motion Planning Trajectory Optimization Stochastic Optimal Control Motion Planning Robot Motion Planning

May 3, 2026

Analytic Bridge Diffusions for Controlled Path Generation

Most modern bridge-diffusion methods achieve finite-time transport by specifying an interpolation, Schrödinger-bridge, or stochastic-control objective and then learning the associated score or drift field with a neural network. In contrast, we identify a restricted but sufficiently broad and analytically solvable class in which the score, intermediate marginals, and protocol gradients are available in closed form without inner stochastic simulation loops and without neural networks in the optimization loop. We recast the classical linear--quadratic--Gaussian (LQG) stochastic-control structure as a transport problem of the Path Integral Diffusion (PID) type. In classical LQG control, linear dynamics, Gaussian noise, and quadratic costs lead to Riccati equations and closed-form optimal feedback. In LQ-GM-PID, we retain the linear--quadratic stochastic-control backbone, but replace terminal state regulation by a prescribed terminal probability density and allow both the initial and terminal laws to be Gaussian Mixtures (GM). Moreover, LQ-GM-PID turns bridge diffusion from a tool for terminal target matching alone into a tool for path shaping. We demonstrate this on a 2D corridor task, a 2D multi-entrance transport task, and a high-dimensional scaling study with

d=32

and

M=16

Gaussian-mixture terminal modes, all with sub-50,ms analytic precompute on a laptop. We position LQ-GM-PID as an analytically solvable reference model for the state-of-the-art neural bridge-diffusion and generative-transport methods: a controlled setting in which neural approximations, score estimates, path-shaping objectives, and protocol-learning procedures can be tested against exact quantities.

Michael Chertkov

Program in Applied Mathematics & Department of Mathematics University of Arizona, AZ 85721, USA

Stochastic Optimal Control Schrödinger Bridges Optimal Control Optimal Transport

Apr 30, 2026

A unified perspective on fine-tuning and sampling with diffusion and flow models

We study the problem of training diffusion and flow generative models to sample from target distributions defined by an exponential tilting of a base density; a formulation that subsumes both sampling from unnormalized densities and reward fine-tuning of pre-trained models. This problem can be approached from a stochastic optimal control (SOC) perspective, using adjoint-based or score matching methods, or from a non-equilibrium thermodynamics perspective. We provide a unified framework encompassing these approaches and make three main contributions: (i) bias-variance decompositions revealing that Adjoint Matching/Sampling and Novel Score Matching have finite gradient variance, while Target and Conditional Score Matching do not; (ii) norm bounds on the lean adjoint ODE that theoretically support the effectiveness of adjoint-based methods; and (iii) adaptations of the CMCD and NETS loss functions, along with novel Crooks and Jarzynski identities, to the exponential tilting setting. We validate our analysis with reward fine-tuning experiments on Stable Diffusion 1.5 and 3.

Carles Domingo-Enrich, Yuanqi Du, Michael S. Albergo

Microsoft Research New England · Harvard University, Kempner Institute

Stochastic Optimal Control Diffusion Sampling Diffusion Model Fine-Tuning Flow-Based Generative Modeling Score-Based Generative Modeling

Apr 30, 2026

Continuous-time q-learning for mean-field control with common noise, part-II: q-learning algorithms

This paper is a continuation work of Ren et al. (2026) aiming to further devise q-learning algorithms for mean-field control (MFC) with controlled common noise. Based on the relaxed control formulation, we first establish the martingale condition of the value function and the Iq-function by evaluating along the conditional state distributions generated by all test policies. As the data in the relaxed control formulation are not observable in practice, we quantify the error incurred when they are replaced by the observable ones in the exploratory formulation under discretely sampled actions. This, together with a two-layer fixed point characterization of an optimal policy in Ren et al. (2026), allows us to propose several algorithms including the Actor-Critic q-learning algorithm, in which the policy is updated in the Actor-step based on the iteration rule induced by the improved Iq-function, and the value function and Iq-function are updated in the Critic-step based on the martingale orthogonality condition using the data from the exploratory formulation. We also establish the convergence of the inner iterations in the Actor-step in an infinite-horizon linear quadratic (LQ) framework. In two examples, within and beyond LQ framework, our q-learning algorithms are implemented with satisfactory performance.

Zhenjie Ren, Xiaoli Wei, Xiang Yu +1

LaMME, Universit´e ´Evry Paris-Saclay, ´Evry, France · Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong · Department of Industrial Engineering and Operations Research, Columbia University, New York, USA

Reinforcement Learning Stochastic Optimal Control Q-Learning Continuous-Time RL Actor-Critic Methods Optimal Control