In recent work it is shown that Q-learning with linear function approximation is stable, in the sense of bounded parameter estimates, under the (ε,κ)-tamed Gibbs policy; κ is inverse temperature, and ε>0 is introduced for additional exploration. Under these assumptions it also follows that there is a solution to the projected Bellman equation (PBE). Left open is uniqueness of the solution, and criteria for convergence outside of the standard tabular or linear MDP settings. The present work extends these results to other variants of Q-learning, and clarifies prior work: a one dimensional example shows that under an oblivious policy for training there may be no solution to the PBE, or multiple solutions, and in each case the algorithm is not stable under oblivious training. The main contribution is that far more structure is required for convergence. An example is presented for which the basis is ideal, in the sense that the true Q-function is in the span of the basis. However, there are two solutions to the PBE under the greedy policy, and hence also for the (ε,κ)-tamed Gibbs policy for all sufficiently small ε>0 and κ≥ 1.

A neural network-based framework is developed and experimentally demonstrated for the problem of estimating the shape of a soft continuum arm (SCA) from noisy measurements of the pose at a finite number of locations along the length of the arm. The neural network takes as input these measurements and produces as output a finite-dimensional approximation of the strain, which is further used to reconstruct the infinite-dimensional smooth posture. This problem is important for various soft robotic applications. It is challenging due to the flexible aspects that lead to the infinite-dimensional reconstruction problem for the continuous posture and strains. Because of this, past solutions to this problem are computationally intensive. The proposed fast smooth reconstruction method is shown to be five orders of magnitude faster while having comparable accuracy. The framework is evaluated on two testbeds: a simulated octopus muscular arm and a physical BR2 pneumatic soft manipulator.

The duality between estimation and control is a foundational concept in control theory. Most students learn about the elementary duality—between observability and controllability—in their first graduate course in linear systems theory. Therefore, it comes as a surprise that for a more general class of nonlinear stochastic systems (HMMs), duality is incomplete. Our objective in writing this article is twofold: 1) to describe the difficulty in extending duality to HMMs and 2) to discuss its recent resolution by the authors. A key message is that the main difficulty in extending duality comes from time reversal when going from estimation to control. The reason for time reversal is explained with the aid of the familiar linear deterministic and linear Gaussian models. The explanation is used to motivate the difference between the linear and the nonlinear models. Once the difference is understood, duality for HMMs is described based on our recent work. The article also includes a comparison and discussion of the different types of duality considered in the literature.

This paper is concerned with the design of algorithms based on systems of interacting particles to represent, approximate, and learn the optimal control law for reinforcement learning (RL). The primary contribution is that convergence rates are greatly accelerated by the interactions between particles. The focus is on the linear quadratic stochastic optimal control problem for which a complete and novel theory is presented. Apart from the new algorithm, sample complexity bounds are obtained, and it is shown that the mean square error scales as 1/N where N is the number of particles. The theoretical results and algorithms are illustrated with numerical experiments and comparisons with other recent approaches, where the faster convergence of the proposed algorithm is numerically demonstrated.

The broad goal of the research surveyed in this article is to develop methods for understanding the aggregate behavior of interconnected dynamical systems, as found in mathematical physics, neuroscience, economics, power systems and neural networks. Questions concern prediction of emergent (often unanticipated) phenomena, methods to formulate distributed control schemes to influence this behavior, and these topics prompt many other questions in the domain of learning. The area of mean field games, pioneered by Peter Caines, are well suited to addressing these topics. The approach is surveyed in the present paper within the context of controlled coupled oscillators.

In the 1940s, Wiener introduced a linear predictor, where the future prediction is computed by linearly combining the past data. A transformer generalizes this idea: it is a nonlinear predictor where the next-token prediction is computed by nonlinearly combining the past tokens. In this essay, we present a probabilistic model that interprets transformer signals as surrogates of conditional measures, and layer operations as fixed-point updates. An explicit form of the fixed-point update is described for the special case when the probabilistic model is a hidden Markov model (HMM). In part, this paper is in an attempt to bridge the classical nonlinear filtering theory with modern inference architectures.

This paper presents a mathematical framework for causal nonlinear prediction in settings where observations are generated from an underlying hidden Markov model (HMM). Both the problem formulation and the proposed solution are motivated by the decoder-only transformer architecture, in which a finite sequence of observations (tokens) is mapped to the conditional probability of the next token. Our objective is not to construct a mathematical model of a transformer. Rather, our interest lies in deriving, from first principles, transformer-like architectures that solve the prediction problem for which the transformer is designed. The proposed framework is based on an original optimal control approach, where the prediction objective (MMSE) is reformulated as an optimal control problem. An analysis of the optimal control problem is presented leading to a fixed-point equation on the space of probability measures. To solve the fixed-point equation, we introduce the dual filter, an iterative algorithm that closely parallels the architecture of decoder-only transformers. These parallels are discussed in detail along with the relationship to prior work on mathematical modeling of transformers as transport on the space of probability measures. Numerical experiments are provided to illustrate the performance of the algorithm using parameter values typical of research-scale transformer models.

This paper is concerned with the error analysis of two types of sampling algorithms, namely model predictive path integral (MPPI) and an interacting particle system (IPS) algorithm, that have been proposed in the literature for numerical approximation of the stochastic optimal control. The analysis is presented through the lens of Gibbs variational principle. For an illustrative example of a single-stage stochastic optimal control problem, analytical expressions for approximation error and scaling laws, with respect to the state dimension and sample size, are derived. The analytical results are illustrated with numerical simulations.

In this article, a biophysically realistic model of a soft octopus arm with internal musculature is presented. The modeling is motivated by experimental observations of sensorimotor control where an arm localizes and reaches a target. Major contributions of this article are: (i) development of models to capture the mechanical properties of arm musculature, the electrical properties of the arm peripheral nervous system (PNS), and the coupling of PNS with muscular contractions; (ii) modeling the arm sensory system, including chemosensing and proprioception; and (iii) algorithms for sensorimotor control, which include a novel feedback neural motor control law for mimicking target-oriented arm reaching motions, and a novel consensus algorithm for solving sensing problems such as locating a food source from local chemical sensory information (exogenous) and arm deformation information (endogenous). Several analytical results, including rest-state characterization and stability properties of the proposed sensing and motor control algorithms, are provided. Numerical simulations demonstrate the efficacy of our approach. Qualitative comparisons against observed arm rest shapes and target-oriented reaching motions are also reported.

Soft continuum arms (SCAs) promise versatile manipulation through mechanical compliance, for assistive devices, agriculture, search applications, or surgery. However, the strong nonlinear coupling between materials, morphology, and actuation renders design and control challenging, hindering real-world deployment. In this context, a modular fabrication strategy paired with reliable, interactive simulations would be highly beneficial, streamlining prototyping and control design. Here, we present a digital twin framework for modular SCAs realized using pneumatic Fiber-Reinforced Elastomeric Enclosures (FREEs). The approach models assemblies of FREE actuators through networks of Cosserat rods, favoring the accurate simulation of three-dimensional arm reconfigurations, while explicitly preserving internal modular architecture. This enables the quantitative analysis and scalable development of composite soft robot arms, overcoming limitations of current monolithic continuum models. To validate the framework, we introduce a three-dimensional reconstruction pipeline tailored to soft, slender, small-volume, and highly deformable structures, allowing reliable recovery of arm kinematics and strain distributions. Experimental results across multiple configurations and actuation regimes demonstrate close agreement with simulations. Finally, we embed the digital twins in a virtual environment to allow interactive control design and sim-to-real deployment, establishing a foundation for principled co-design and remote operation of modular soft continuum manipulators.

This paper is divided into two parts. The first part reviews the formulae for f-divergences in the study of continuous-time Markov processes and explores their applications in areas such as stochastic stability, the second law of thermodynamics, and its non-equilibrium extensions. This sets the foundation for the second part, which focuses on f-divergence in the study of hidden Markov processes. In this context, we present analyses of filter stability and stochastic thermodynamics, with the latter being used to illustrate the concept of a Maxwell demon in an over-damped Langevin model with white noise observations. The paper's expository style and unified formalism for both Markov and hidden Markov processes aim to serve as a valuable resource for researchers working across related fields.

Muscular hydrostats, such as octopus arms or elephant trunks, lack bones entirely, endowing them with exceptional dexterity and reconfigurability. Key to their unmatched ability to control nearly infinite degrees of freedom is the architecture into which muscle fibers are weaved. Their arrangement is, effectively, the instantiation of a sophisticated mechanical program that mediates, and likely facilitates, the control and realization of complex, dynamic morphological reconfigurations. Here, by combining medical imaging, biomechanical data, live behavioral experiments, and numerical simulations, an octopus-inspired arm made of [Formula: see text]200 continuous muscle groups is synthesized, exposing "mechanically intelligent" design and control principles broadly pertinent to dynamics and robotics. Such principles are mathematically understood in terms of storage, transport, and conversion of topological quantities, effected into complex 3D motions via simple muscle activation templates. These are in turn composed into higher-level control strategies that, compounded by the arm's compliance, are demonstrated across challenging manipulation tasks, revealing surprising simplicity and robustness.

In this paper, stochastic optimal control problems in continuous time and space are considered. In recent years, such problems have received renewed attention from the lens of reinforcement learning (RL) which is also one of our motivation. The main contribution is a simulation-based algorithm -- dual ensemble Kalman filter (EnKF) -- to numerically approximate the solution of these problems. The paper extends our previous work where the dual EnKF was applied in deterministic settings of the problem. The theoretical results and algorithms are illustrated with numerical experiments.

This paper is concerned with the development and use of duality theory for a nonlinear filtering model with white noise observations. The main contribution of this paper is to introduce a stochastic optimal control problem as a dual to the nonlinear filtering problem. The mathematical statement of the dual relationship between the two problems is given in the form of a duality principle. The constraint for the optimal control problem is the backward stochastic differential equation (BSDE) introduced in the companion paper. The optimal control solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The proposed duality is shown to be an exact extension of the classical Kalman-Bucy duality, and different from other types of optimal control and variational formulations given in literature.

This paper is concerned with the problem of nonlinear (stochastic) filter stability of a hidden Markov model (HMM) with white noise observations. The main contribution is the variance decay property which is used to conclude filter stability. The property is closely inspired by the Poincaré inequality (PI) in the study of stochastic stability of Markov processes. In this paper, the property is related to both the ergodicity of the Markov process as well as the observability of the HMM. The proofs are based upon a recently discovered minimum variance duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE).

ABSTRACT In this report, passive elasticity properties of Octopus rubescens arm tissue are investigated using a multidisciplinary approach encompassing biomechanical experiments, computational modeling, and analyses. Tensile tests are conducted to obtain stress–strain relationships of the arm under axial stretch. Rheological tests are also performed to probe the dynamic shear response of the arm tissue. Based on these tests, comparisons against three different viscoelasticity models are reported.

A backward map is introduced for the purposes of analysis of nonlinear (stochastic) filter stability. The backward map is important because filter-stability, in the sense of chi(2)divergence, follows from a certain variance decay property associated with the backward map. To show this property requires additional assumptions on the hidden Markov model (HMM). The analysis in this paper is based on introducing a Poincare Inequality (PI) for HMMs with white noise observations. In finite state-space settings, PI is related to both the ergodicity of the Markov process as well as the observability of the HMM. It is shown that the Poincare constant is positive if and only if the HMM is detectable.

This paper is in two parts. In the first part of this paper, formulae for f-divergence for a continuous Markov process are reviewed. Applications of these formulae are described for the problems of stochastic stability, second law of thermodynamics, and non-equilibrium extensions thereof. The first part sets the stage for considering the f-divergence for hidden Markov processes which is the focus of the second part. For hidden Markov processes, again based on the use of f-divergence, analyses of filter stability and stochastic thermodynamics are described. The latter is used to illustrate the Maxwell demon for an over-damped Langevin model with white noise type observations. The expository nature of the paper combined with consistent formalism for both Markov and hidden Markov processes is written to be useful for researchers working on related themes in disparate areas.

This paper is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation (BSDE) as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulae, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table~II).