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.

This paper addresses boundary control to input-to-state stabilization for fractional convection-diffusion-reaction (FCDR) systems governed by coupled time fractional partial differential equations (TFPDEs) under matched and unmatched disturbances over actuator/sensor networks using output feedback, fractional sliding mode (FSM) algorithm and sampled-in-space sensing. Here it is assumed that sensors provide discrete in space measurements, i.e., spatially averaged measurements (SAMs), and a limited number of sensors are allocated in a spatial domain. A sampled-data observation problem is first in investigation, which contains to design a FSM observer against boundary disturbances and to prove input-to-state stability (ISS) of the proposed observer. Using this observer and backstepping approach, we develop an output feedback FSM controller and establish the reaching condition to FSM surface. Using the fractional Lyapunov method, ISS of the closed-loop dynamics is achieved. Theoretical results are verified by numerical simulations.

The authors study an approximation method for partially observed Markov decision processes (POMDPs) with continuous spaces. Belief MDP reduction, which has been the standard approach to study POMDPs requires rigorous approximation methods for practical applications, due to the state space being lifted to the space of probability measures. Generalizing recent work, in this paper the authors present rigorous approximation methods via discretizing the observation space and constructing a fully observed finite MDP model using a finite length history of the discrete observations and control actions. The authors show that the resulting policy is near-optimal under some regularity assumptions on the channel, and under certain controlled filter stability requirements for the hidden state process. The authors also provide a Q learning algorithm that uses a finite memory of discretized information variables, and prove its convergence to the optimality equation of the finite fully observed MDP constructed using the approximation method.

This paper considers the practical fixed-time tracking control problem for a state constrained pure-feedback nonlinear system. A new barrier function is first proposed to handle various asymmetric time-varying constraints and unify the cases with and without state constraints. Then a low-cost neural network based adaptive fixed-time controller is constructed by further combining the dynamic surface control, which overcomes the technical problems of overparametrization and singularity in the backstepping procedure. Our design guarantees that the tracking error converges to a small neighbourhood of zero in a fixed time while satisfying the state constraints as a priority task without imposing feasibility conditions on the virtual controllers. Simulation results validate the effectiveness of the proposed adaptive fixed-time tracking control strategy.

This paper considers risk-sensitive linear-quadratic mean-field games. By the so-called direct approach via dynamic programming, we determine the feedback Nash equilibrium in an $N$-player game. Subsequently, we design a set of decentralized strategies by passing to the mean-field limit. We prove that the set of decentralized strategies constitutes an $O(1/N)$-Nash equilibrium when applied by the $N$ players, and hence obtain so far the tightest equilibrium error bounds for this class of models.

In this paper, the authors revisit decentralized control of linear quadratic (LQ) systems. Instead of imposing an assumption that the process and observation noises are Gaussian, the authors assume that the controllers are restricted to be linear. The authors show that the multiple decentralized control models, the form of the best linear controllers is identical to the optimal controllers obtained under the Gaussian noise assumption. The main contribution of the paper is the solution technique. Traditionally, optimal controllers for decentralized LQ systems are identified using dynamic programming, maximum principle, or spectral decomposition. The authors present an alternative approach which is based by combining elementary building blocks from linear systems, namely, completion of squares, state splitting, static reduction, orthogonal projection, (conditional) independence of state processes, and decentralized estimation.

Optimal feedback control of nonlinear system with free terminal time present many challenges including nonsmooth in the value function and control laws, and existence of multiple local or even global optimal trajectories. To mitigate these issues, the authors introduce an actor-critic method along with some enhancements. The authors demonstrate the algorithm's effectiveness on a prototypical example featuring each of the main pathological issues present in problems of this type as well as a higher dimensional example to show that the solution method presented can scale.

This study aims to solve the Nash equilibrium (NE) seeking problem for monotone $N$-coalition games. The authors assume that the gradient mapping of the game is monotone but not necessarily strictly or strongly monotone. Such a merely monotone assumption presents significant challenges to NE seeking, since the basic gradient descent method may fail to converge. The authors start with a regularization-based projected gradient dynamical system in a general non-cooperative game framework and analyze the convergence of the dynamics under different scenarios. Then, the authors develop NE seeking algorithms for monotone $N$-coalition games with undirected and connected inner-coalition communication graphs. Asymptotic convergence to the least-norm NE is proven. The convergence rate of the algorithm for an analytic mapping is provided. Furthermore, the authors propose a novel regularization-based dynamical system that allows different parameters among the coalitions. Rigorous analysis and a numerical example are provided to illustrate the effectiveness of the proposed method.

In this paper the authors consider the operational problem of optimal signalling and control, called control-coding capacity (with feedback), $C_{FB}$ in bits/second, of discrete-time nonlinear partially observable stochastic systems in state space form, subject to an average cost constraint. $C_{FB}$ is the maximum rate of encoding signals or messages into randomized controller-encoder strategies with feedback, which control the state of the system, and reproducing the messages at the output of the system using a decoder or estimator with arbitrary small asymptotic error probability. In the first part of the paper, the authors characterize $C_{FB}$ by an information theoretic optimization problem of maximizing directed information from the inputs to the outputs of the system, over randomized strategies (controller-encoders). The authors derive equivalent characterizations of $C_{FB}$, using randomized strategies generated by either uniform or arbitrary distributed random variables (RVs), sufficient statistics, and a posteriori distributions of nonlinear filtering theory. In the second part of the paper, the authors analyze $C_{FB}$ for linear-quadratic Gaussian partially observable stochastic systems (LQG-POSSs). The authors show that randomized strategies consist of control, estimation and signalling parts, and the sufficient statistics are, two Kalman-filters and an orthogonal innovations process. The authors prove a semi-separation principle which states, the optimal control strategy is determined explicitly from the solution of a control matrix difference Riccati equation (DRE), independently of the estimation and signalling strategies. Finally, the authors express the optimization problem of $C_{FB}$ in terms of two filtering matrix DREs, a control matrix DRE, and the covariance of the innovations process. Throughout the paper, the authors illustrate that the expression of $C_{FB}$ includes as degenerate cases, problems of stochastic optimal control and channel capacity of information transmission.

This paper considers the value iteration algorithms of stochastic zero-sum linear quadratic games with unkown dynamics. On-policy and off-policy learning algorithms are developed to solve the stochastic zero-sum games, where the system dynamics is not required. By analyzing the value function iterations, the convergence of the model-based algorithm is shown. The equivalence of several types of value iteration algorithms is established. The effectiveness of model-free algorithms is demonstrated by a numerical example.

This paper proposes a data-driven learning-based approach to predictive control for switched nonlinear systems subject to state and control constraints and external stochastic disturbances. A switched Koopman modeling framework is developed, where a multi-mode neural network for state lifting is trained simultaneously with Koopman operators and state reconstruction matrices for all the modes. This framework facilitates the construction of a switched linear Koopman model in a transformed space and effectively captures the dynamics of the original nonlinear system. A switched predictive control strategy is then designed to regulate the switched Koopman model with constrained state and control inputs against both the stochastic disturbances and the uncertainties introduced by the lifting neural network. The proposed control scheme ensures mean-square stability and guarantees boundedness during the online phase. Furthermore, boundedness analysis is performed to determine the reachable set of the original system state across all admissible switching sequences. The effectiveness of the proposed methodology is demonstrated through a case study of a gene regulatory network.

This paper focuses on solving the distributed optimization problem with binary-valued intermittent measurements of local objective functions. In this context, a binary-valued measurement represents whether the measured value is smaller than a fixed threshold. Meanwhile, the "intermittent|" scenario arises when there is a non-zero probability of not detecting each local function value during the measuring process. Using this kind of coarse measurement, we propose a discrete-time stochastic extremum seeking-based algorithm for distributed optimization over a directed graph. As is well-known, many existing distributed optimization algorithms require a doubly-stochastic weight matrix to ensure the average consensus of agents. However, in practical engineering, achieving double-stochasticity, especially for directed graphs, is not always feasible or desirable. To overcome this limitation, we design a row-stochastic matrix and a column-stochastic matrix as weight matrices in our algorithm instead of relying on doubly-stochasticity. Under some mild conditions, we rigorously prove that agents can reach the average consensus and ultimately find the optimal solution. Finally, we provide a numerical example to illustrate the effectiveness of our algorithm.

In this paper, we consider a wave-plate transmission system on Riemannian manifold with boundary controls on the plate equation. This model arises from the noise control and is extended to the broader context of the Riemannian manifold. By employing the semigroup method, we obtain the well-posedness of the system. Subsequently, under certain geometric conditions, we establish a polynomial energy decay estimate of type $t^{-1}$ by using the geometric multiplier method in which the conclusion proposed by Guo and Yao (J. Math. Anal. Appl. 2006, 317: 50-70.) plays a crucial role in our proof. It effectively addresses the challenge posed by curvature on the Riemannian manifold. Moreover, by employing the higher-order energy method, we overcome the technical difficulties caused by higher-order terms in boundary controls, which generalizes the results in the literature.

In this paper, we consider a class of linear quadratic optimal control problems of backward stochastic differential equations under partial information with initial value constraint. The main contribution is to obtain an explicitly optimal controller by using a Riccati equation and an optimal parameter characterized by a matrix equation. The key technique is to introduce Lagrange multiplier to transform the problem with initial value constraint into unconstrained optimal control problem and optimal parameter calculation problem, and solve the optimal parameter calculation by using the exact controllability of the system.

This paper addresses an indefinite linear-quadratic mean field games for stochastic large-population system, where the individual diffusion coefficients may depend on both the state and the control of the agents. Moreover, the control weights in the cost functionals could be indefinite. We employ a direct approach to derive the $\epsilon$-Nash equilibrium strategy. First, we formally solve an $N$-player game problem within a vast but finite population setting. Subsequently, by introducing two Riccati equations, we decouple or reduce the high-dimensional systems to yield centralized strategies, which depend on the state of a specific player and the average state of the population. As the population size $N$ goes infinity, the construction of decentralized strategies becomes feasible. Then, we demonstrate that these strategies constitute an $\epsilon$-Nash equilibrium. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed strategies.

In this paper, the issues of both reconstructibility analysis and state estimation are investigated for locally reconstructible Boolean networks (BNs), where the pinning control is employed to design a special control sequence. First, this paper will focus on dealing with state unreconstructible cycle based on state set-estimation theory, and the concept of set-to-set reachability is proposed. Second, the locally reconstructible BNs are transformed to Boolean control networks (BCNs) by pinning control technique, where a logical matrix is designed by a special algorithm such that all states in the unreconstructible initial state set can evolve into the reconstructible state set, simultaneously. Third, a sufficient condition for the existence of a free Boolean control sequence is obtained, which guarantees that a locally reconstructible BN can be transformed into a reconstructible one. Besides, an optimization control algorithm is proposed to generate pinning control sequence and further optimize the reconstructibility of BNs. A kind of pinning control-based state estimation method is also developed. Finally, an example is provided to show the feasibility of the proposed methods.

This paper investigates an innovative predefined time output feedback adaptive fuzzy consensus control method for fractional-order nonlinear multi-agent systems(FONMASs). The controlled system includes unmeasurable states and unknown nonlinear functions. The fuzzy state observer is built to evaluate unmeasurable state variables, unknown nonlinear functions are addressed through fuzzy logic system approximation. Combining with fractional-order dynamic surface control (DSC) and the theory of stability of predefined time, the control method is raised. By employing FODSC techniques, the computational explosion problem can be avoided. The stability of the closed-loop system is analyzed using the Lyapunov function, demonstrating that FONMAS achieves semi-global practical predefined time stability (SGPPTS). Eventually, the validity of the prescribed control strategy is verified by a simulation case.

This paper studies a distributed policy gradient in collaborative multi-agent reinforcement learning (MARL), where agents communicating over a network aim to find an optimal policy that maximizes the average of all the agents' local returns. To address the challenges of high variance and bias in stochastic policy gradients for MARL, this paper proposes a distributed policy gradient method with variance reduction, combined with gradient tracking to correct the bias resulting from the difference between local and global gradients. We also utilize importance sampling to solve the distribution shift problem in the sampling process. We then show that the proposed algorithm finds an $\epsilon$-approximate stationary point, where the convergence depends on the number of iterations, the mini-batch size, the epoch size, the problem parameters, and the network topology. We further establish the sample and communication complexity to obtain an $\epsilon$-approximate stationary point. Finally, numerical experiments are performed to validate the effectiveness of the proposed algorithm.

This paper investigates the path-guided distributed formation control of networked autonomous surface vehicles (ASVs) subject to model uncertainties and environmental disturbances. A safety-certified path-guided coordinated control method is proposed for multiple ASVs to achieve a distributed formation in obstacle environments. Specifically, a neural predictor with a high-order tuner is presented to approximate unknown nonlinearities with accelerated learning performance. Subsequently, control Lyapunov functions (CLFs) and control barrier functions (CBFs) are constructed for mapping stability constraints and safety constraints on states to control inputs. A quadratic optimization problem is constructed with the norm of control inputs as the objective function, CLFs and CBFs as constraints. Neurodynamic optimization is used to deal with the quadratic programming problem and generate the optimal kinetic control signals, thereby attaining the desired safe formation. Unlike the high-order CBF, a CBF backstepping method is proposed to establish safety constraints such that repeated time derivatives of system nonlinearities can be avoided. The multi-ASVs system is ensured to be input-to-state safe irrespective of high-order relative degree. Through the Lyapunov theory, the multi-ASVs system is proven to be input-to-state stable. Finally, simulation results are presented to validate the efficacy of the presented safety-certified distributed formation control for networked ASVs.

The distributed optimal output synchronization problem for the leaderless heterogeneous multi-agent system with a general global cost function is investigated for the first time by linear quadratic (LQ) optimal control theory. Conventional algorithms for heterogeneous systems are quite complex, requiring the design of a virtual reference generator and the solving of regulation equations. This paper presents a novel distributed asymptotically optimal controller by incorporating the design of distributed observer and feedforward controller. A general form of the distributed controller is obtained by solving an augmented algebraic Riccati equation, which is parallel to classical optimal control theory. The optimal topology is an arbitrary directed graph containing only a spanning tree. It is shown that the proposed algorithms outperform the traditional consensus methods in the convergence speed by selecting proper observer gain matrices, and eliminate the reliance on the nonzero eigenvalues of Laplacian matrix. Simulation example further demonstrates the effectiveness of the proposed scheme and a faster superlinear convergence speed than the existing algorithm.

An algorithm for computing parametric order bases for univariate polynomial matrices with parameters is first presented in this paper. Starting from the non-parametric univariate polynomial matrix, our key idea is to construct a special module and module order. Then based on Gröbasis theory for modules, we present that the order basis can be obtained by computing a minimal Gröbasis for this module under this order. Further, we extend the definition of the order basis to the parametric polynomial matrix, and give the concept of comprehensive order basis systems. More importantly, the method based on Gröbases for modules can be naturally generalized to the parametric case by means of comprehensive Grösystems for modules. As a consequence, we design a new algorithm for computing comprehensive order basis systems. The proposed algorithm has been implemented on the computer algebra system Singular and Maple.

This paper considers the real-time estimation problem of vehicle mass, which has a significant impact on driving comfort and safety. A bilinear parameter identification algorithm is proposed for a type of nonlinear identification problems, which encompass vehicle mass estimation. The feature of this nonlinear model is that two parameters to be estimated are multiplied together, which brings great difficulties to identification compared to linear models. The main idea proposed in the algorithm design is to transform the original nonlinear model into two mutually dependent linear models, which are identified by the recursive algorithms. By constructing a combined Lyapunov function, it is theoretically proved that the algorithm converges under the input excitation condition, and the convergence rate $O(1/t)$ is achieved based on some extra mild conditions. Finally, the algorithm is verified through practical experiments, with the estimated vehicle mass error of $1.06%$ on average, which shows the feasibility of the algorithm.

Feedback optimization aims at regulating the output of a dynamical system to a value that minimizes a cost function. This problem is beyond the reach of the traditional output regulation theory, because the desired value is generally unknown and the reference signal evolves according to a gradient flow using the system's real-time output. This paper complements the output regulation theory with the nonlinear small-gain theory to address this challenge. Specifically, the authors assume that the cost function is strongly convex and the nonlinear dynamical system is in lower triangular form and is subject to parametric uncertainties and a class of external disturbances. An internal model is used to compensate for the effects of the disturbances while the cyclic small-gain theorem is invoked to address the coupling between the reference signal, the compensators, and the physical system. The proposed solution can guarantee the boundedness of the closed-loop signals and regulate the output of the system towards the desired minimizer in a global sense.Two numerical examples illustrate the effectiveness of the proposed method.

Rosenblatt and Rosenblatt-Volterra processes are two families of stochastic processes that are described by double Wiener-Itô integrals with singular kernels. The Rosenblatt processes have exponential singular kernels and the Rosenblatt-Volterra processes have singular Volterra kernels for the Wiener-Itô integrals. Empirical evidence shows that for many control systems the assumption of Gaussian noise is not appropriate so Rosenblatt and Rosenblatt-Volterra processes are some generalizations of Gaussian processes that can provide natural alternatives to Gaussian probability laws. Furthermore, the results for Rosenblatt and Rosenblatt-Volterra processes are tractable for some applications. These results can be compared to prediction for Gaussian processes and Gauss-Volterra processes.

In econometric analysis, researchers often encounter vast datasets that greatly reduce model efficiency. In this paper, the authors develop a sequential shrinkage estimation method for use in distributed settings. Within this framework, one dataset is split into several blocks, and each block is regarded as a node. The sequential shrinkage estimation method is implemented on the data in each block until the stopping criteria are satisfied. These sequential procedures are then integrated to produce the final results using a weighted average, which provides approximate regression result estimates for the entire dataset. The proposed method can significantly reduce the required number of samples and perform parameter estimation and variable selection while satisfying the preset accuracy requirements. In addition, the statistical properties of the parameter estimates in the proposed approach are analyzed for use in a linear regression model. Finally, numerical studies on simulated and real datasets show that the proposed method performs well.

In this paper, the authors propose an approach to properly aggregate a reversible discrete-time switched linear system and prove that, for any $n$-dimensional exponentially stabilizable switched system, the authors could design up to $n$ linear gain matrices, such that the extended system is also exponentially stabilizable as a switched autonomous system. By utilizing the pathwise state feedback switching strategy of the switched autonomous system, the original system is aggregated into a piecewise linear system that is step-wise norm contractive and exponentially stable. The authors also develop a robust switching design mechanism that simultaneously achieves exponential stability, structural stability, and input-to-state stability for the closed-loop system. A numerical example is presented to demonstrate the effectiveness of the proposed design scheme.

Building heating, ventilating, and air conditioning (HVAC) systems have one of the largest energy footprint worldwide, which necessitates the design of intelligent control algorithms that improve the energy utilization while still providing thermal comfort. In this work, the authors formulate the HVAC equipment dynamics in the setting of a two-player non-zero-sum cooperative game, which enables two decision variables (mass flow rate and supply air temperature) to perform joint optimization of the control utilization and thermal setpoint tracking by simultaneously exchanging their policies. The HVAC zone serves as a game environment for these two decision variables that act as two players in a game. It is assumed that dynamic models of HVAC equipment are not available. Furthermore, neither the state nor any estimates of HVAC disturbance (heat gains, outside variations, etc.) are accessible, but only the measurement of the zone temperature is available for feedback. Under these constraints, the authors develop a new data-driven Q-learning scheme employing policy iteration and value iteration with a bias compensation mechanism that accounts for unmeasurable disturbances and circumvents the need of full-state measurement. The proposed algorithms are shown to converge to the optimal solution corresponding to the generalized algebraic Riccati equations (GAREs) in dynamic games.

This paper presents a dynamic game framework to analyze the role of large banks in interbank markets. By extending existing models, the authors incorporate a major bank as a dynamic decision-maker interacting with multiple small banks. Using the mean-field game methodology and convex analysis, best-response trading strategies are derived, leading to an approximate equilibrium for the interbank market. The authors investigate the influence of the large bank on the market stability by examining individual default probabilities and systemic risk, through the use of Monte Carlo simulations. The proposed findings reveal that, when the size of the major bank is not excessively large, it can positively contribute to market stability. However, there is also the potential for negative spillover effects in the event of default, leading to an increase in systemic risk. The magnitude of this impact is further influenced by the size and trading rate of the major bank. Overall, this study provides valuable insights into the management of systemic risk in interbank markets.

In the analysis of censored survival data, it is crucial to consider the heterogeneity of treatment effects in order to avoid biased inferences. The deep neural network-based heterogeneous partially linear Cox model offers a flexible and robust solution by incorporating both heterogeneous linear and homogeneous nonlinear components. We propose a novel approach to subgroup detection for this model under right-censoring, using deep neural networks to approximate nonlinear effects. To simultaneously estimate parameters and identify subgroups, we employ a concave pairwise penalty and the alternating direction method of multipliers (ADMM) algorithm. Furthermore, we demonstrate that the proposed estimator possesses oracle properties and achieves model selection consistency. Through simulation studies and empirical data analysis on breast cancer, we illustrate the effectiveness of our proposed method.

This paper investigates the multi-player non-zero-sum game problem for unknown linear continuous-time systems with unmeasurable states. By only accessing the data information of input and output, a data-driven learning control approach is proposed to estimate $N$-tuple dynamic output feedback control policies which can form Nash equilibrium solution to the multi-player non-zero-sum game problem. In particular, the explicit form of dynamic output feedback Nash strategy is constructed by embedding the internal dynamics and solving coupled algebraic Riccati equations. The coupled policy-iteration based iterative learning equations are established to estimate the $N$-tuple feedback control gains without prior knowledge of system matrices. Finally, an example is used to illustrate the effectiveness of the proposed approach.

This article formulates interactive adversarial differential graphical games for synchronization control of multiagent systems (MASs) subject to adversarial inputs interacting with the systems through topology communications. Local control and interactive adversarial inputs affect each agent's local synchronization error via local networks. The distributed global Nash equilibrium (NE) solutions are guaranteed in the games by solving the optimal control input of each agent and the worst-case adversarial input based solely on local states and communications. The asymptotic stability of the local synchronization error dynamics and the NE are guaranteed. Furthermore, the authors devise a data-driven online reinforcement learning (RL) algorithm that only computes the distributed Nash control online using system trajectory data, eliminating the need for explicit system dynamics. A simulation-based example validates the game and algorithm.

This paper presents a global robust nonlinear output regulation (GROR) design for nonlinear systems that do not necessarily exhibit hyperbolic zero dynamics. The hyperbolic condition has been predominantly required in existing literature on GROR, particularly for smooth global asymptotic stabilization in various scenarios. This limitation has motivated the current investigation to relevant global regulation control problems. Building on the paradigm of "reduction of the plant dynamics and augmentation of the exosystem" (termed Reduction-Augmentation) proposed in [？], we develop an internal model-based Lyapunov approach to achieving GROR through smooth error-output feedback under mild conditions. Notably, we establish a smooth global stabilizer by means of a Lyapunov's direct method for the augmented system using the tool of input-to-state stability (ISS) with respect to a compact zero-invariant set. As an interesting outcome, the proposed method applies to nonlinear systems under strictly relaxed conditions than previous studies.

This paper presents in organized form a number of results that have appeared in the literature in the last two decades, concerning the design of control laws for multi-input multi-output nonlinear systems, with emphasis on the problem of stabilizing an equilibrium, and addresses, at a broad level generality, systems that are invertible from an input-output viewpoint.

This paper explores the fundamental question of the degrees of controllability (in the sense of Kalman-Friedland) of Keynesian-type linear multiplier processes of income-import-primary input changes for a significant class of actual input-output table economies based on a dynamic and extended version of the Kurz multi-sector multiplier model and providing relevant spectral evidence from extensive input-output table data. The findings suggest that, irrespective of the direction of the control-policy instrument vector, (i) the degrees of controllability are not greater than the order of $10^{-19}$; (ii) for a tolerance of $10^{-4}$ (of $10^{-2}$), the normalised numerical rank of the controllability matrices is not greater than 17% (than 10%); (iii) the largest ‘spreads’ between two consecutive singular values of the controllability matrices are not greater than $10^{-1}$ and, therefore, (iv) the dynamic multiplier systems are ‘structurally almost uncontrollable’. Since these controllability characteristics arise from the underlying-deep production-consumption structure of the considered class of actual input-output table economies, and since the effectiveness of demand management policies is at the centre of controversy in the post-2008 crisis world, the findings of this paper are of interest for both theoretical and applied policy studies.

Nonlinear feedback shift registers (NFSRs) are used in many stream ciphers as their main building blocks. An NFSR is said to be observable if any two distinct initial states are distinguishable from their resulting output sequences, and it is said to be nonsingular if its state diagram contains only cycles. To avoid weak key attacks, stream ciphers must use nonsingular and observable NFSRs. With condition that NFSRs only output their first bits, Fibonacci NFSRs are clearly observable, but Galois NFSRs' observability is more complex and is little studied. This paper continues the research on the observability of Galois NFSRs with relaxation of their output restriction. Resorting to the semi-tensor product based Boolean network theory, the paper first reveals the observability index's lower bound for general Galois NFSRs and its upper bound for singular and nonsingular Galois NFSRs and further for nonsingular (sub)maximum-cycle Galois NFSRs. It then gives some necessary and/or sufficient conditions for the observability of general Galois NFSRs. Based on the disclosure of the characterizations of distinguishable initial states on a cycle, the paper finally provides some necessary and/or sufficient conditions for the observability of nonsingular (sub)maximum-cycle Galois NFSRs, helpful to the design of stream ciphers.

With the development and applications of the Smart Court System (SCS) in China, the reliability and accuracy of legal artificial intelligence have become focal points in recent years. Notably, criminal sentencing prediction, a significant component of the SCS, has also garnered widespread attention. According to the Chinese criminal law, actual sentencing data exhibits a saturated property due to statutory penalty ranges, but this mechanism has been ignored by most existing studies. Given this, we propose a sentencing prediction model that combines judicial sentencing mechanisms including saturated outputs and floating boundaries with neural networks. Building on the saturated structure of our model, a more effective adaptive prediction algorithm will be constructed based on the fusion of several key ideas and techniques that include the utilization of the $L_1$ loss together with the corresponding gradient update strategy, a data pre-processing method based on large language model to extract semantically complex sentencing elements using prior legal knowledge, the choice of appropriate initial conditions for the learning algorithm and the construction of a double-hidden-layer network structure. An empirical study on the crime of disguising or concealing proceeds of crime demonstrates that our method can achieve superior sentencing prediction accuracy and significantly outperform common baseline methods.

We provide a method for determining liouvillian solutions of a class of first-order algebraic ordinary differential equations (AODEs). In more details, we separate the method in two parts. if a first-order AODE is of genus zero, we prove that its liouvillian solutions can be found via a class of quasi linear first-order ordinary differential equations by means of associated field of algebraic functions and rational parametrizations. In addition, we propose an algorithm to determine the reduced form of a certain first-order AODE by means of power transformations. This induces a novel approach for finding liouvillian solutions of first-order AODEs of positive genera whose reduced form are of genus zero. Some examples are given to illustrate the method in the affirmative cases.

The authors consider the property of detectability of discrete event systems in the presence of sensor attacks in the context of cyber-security. The authors model the system using an automaton and study the general notion of detectability where a given set of state pairs needs to be (eventually or periodically) distinguished in any estimate of the state of the system. The authors adopt the ALTER sensor attack model from previous work and formulate four notions of CA-detectability in the context of this attack model based on the following attributes: strong or weak; eventual or periodic. The authors present verification methods for strong CA-detectability and weak CA-detectability. The authors present definitions of strong and weak periodic CA-detectability that are based on the construction of a verifier automaton called the augmented CA-observer. The development also resulted in relaxing assumptions in prior results on D-detectability, which is a special case of CA-detectability.

Evanescent random walks are instances of stochastic processes that terminate at a specific rate. They have proved relevant in modeling diverse behaviors of complex systems from protein degradation in gene networks (Ali and Brewster (2022), Ghusinga, et al. (2017), and Ham, et al. (2024)) and "nonprocessive" motor proteins (Kolomeisky and Fisher (2007)) to decay of diffusive radioactive matter (Zoia (2008)). The present work aims to extend a well-established estimation and control problem, the so-called Schrödinger's bridge problem, to evanescent diffusion processes. Specifically, the authors seek the most likely law on the path space that restores consistency with two marginal densities—One is the initial probability density of the flow, and the other is a density of killed particles. The Schrödinger's bridge problem can be interpreted as an estimation problem but also as a control problem to steer the stochastic particles so as to match specified marginals. The focus of previous work in Eldesoukey, et al. (2024) has been to tackle Schrödinger's problem involving a constraint on the spatio-temporal density of killed particles, which the authors revisit here. The authors then expand on two related problems that instead separately constrain the temporal and the spatial marginal densities of killed particles. The authors derive corresponding Schrödinger systems that contain coupled partial differential equations that solve such problems. The authors also discuss Fortet-Sinkhorn-like algorithms that can be used to construct the sought bridges numerically.

The authors consider the problem of reaching consensus over a communication network via asynchronous interaction between pairs of agents. A well-known method is the linear gossip algorithm due to Tsitsiklis (1984). Extension of this, allowing the selection of a strictly stationary sequence of communicating pairs, was given in Picci and Taylor (2013). Extension of the linear gossip algorithm to directed communication networks, retaining the linear dynamics, was proposed by Cai and Ishii (2012), later extended by Silvestre, et al. (2018). A definite novelty of these algorithms is that $L_2$-convergence with exponential rate can be established. The authors attend the above issues, extending the result of Picci and Taylor (2013) motivated by features of algorithms for directed networks. The authors present and discuss the algorithm of Silvestre, et al. (2018), together with systematic simulation results based on 5M randomly chosen parameter settings. The core of the proposed mathematical technology is a set of simple observations, presented with a tutorial aspect, by which the authors can conveniently establish various results on the almost sure convergence of products of strictly stationary sequences of matrices to a rank-1 matrix.