Discrete-time Setting Research Articles

Large deviations principle via Malliavin calculus for the Navier–Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker–Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier–Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in [17] in a discrete-time setting and extended in [26] to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman–Kac semigroup, which we verify by using Malliavin calculus.

Precise Dynamic Consensus under Event-Triggered Communication

This work addresses the problem of dynamic consensus, which consists of estimating the dynamic average of a set of time-varying signals distributed across a communication network of multiple agents. This problem has many applications in robotics, with formation control and target tracking being some of the most prominent ones. In this work, we propose a consensus algorithm to estimate the dynamic average in a distributed fashion, where discrete sampling and event-triggered communication are adopted to reduce the communication burden. Compared to other linear methods in the state of the art, our proposal can obtain exact convergence under continuous communication even when the dynamic average signal is persistently varying. Contrary to other sliding-mode approaches, our method reduces chattering in the discrete-time setting. The proposal is based on the discretization of established exact dynamic consensus results that use high-order sliding modes. The convergence of the protocol is verified through formal analysis, based on homogeneity properties, as well as through several numerical experiments. Concretely, we numerically show that an advantageous trade-off exists between the maximum steady-state consensus error and the communication rate. As a result, our proposal can outperform other state-of-the-art approaches, even when event-triggered communication is used in our protocol.

A two‐stage method for the shipper lane selection problem with time windows in transportation service procurement

AbstractThe shipper lane selection problem (SLSP) is determining which lanes are to be served by the shipper's vehicle fleet or outsourced to the carriers through auction. While the SLSP in previous studies assumes that each lane is associated with a set of discrete times at which it can be served, this study considers a generalized version assuming a lane service time window for each lane, which is named the SLSP with time windows (SLSPTW). The SLSPTW is formulated as a mixed integer linear programming model that minimizes the sum of the transportation and the service/setup costs incurred by the shipper to auction off the lanes served by the carriers. In the proposed two‐stage solution approach, the first stage is designed to generate quickly a set of possible solutions, the best of which is then verified by a decomposed model of the SLSPTW at the second stage. Besides, an iterative‐improvement mechanism for the proposed algorithm is adopted to achieve the efficiency of the solution quality improvement. The effectiveness and efficiency of the developed solution method is demonstrated by the conducted numerical experiments.

Fixed-time parameter estimation via the discrete-time DREM method

A simple fixed-time converging estimation algorithm is presented for a linear regression using the dynamic regressor extension and mixing (DREM) method within a discrete-time setting, with a persistently exciting regressor and bounded measurement noises. The solution is based on Kreisselmeier's filters and is computationally simpler than the existing analogs.

Valuation of cliquet-style guarantees with death benefits

<p style='text-indent:20px;'>In this paper, we consider the problem of valuing an equity-linked insurance product with a cliquet-style payoff. The premium is invested in a reference asset whose dynamic is modeled by a geometric Brownian motion. The policy delivers a payment to the beneficiary at either a fixed maturity or the time upon the insured's death, whichever comes first. The residual lifetime of a policyholder is described by a random variable, assumed to be independent of the asset price process, and its distribution is approximated by a linear sum of exponential distributions. Under such characterization, closed-form valuation formulae are derived for the contract considered. Moreover, a discrete-time setting is briefly discussed. Finally, numerical examples are provided to illustrate our proposed approach.</p>

Systematic Design of Discrete-Time Control Barrier Functions Using Maximal Output Admissible Sets

This paper introduces a novel discrete-time control barrier function (DCBF) that stems directly from discrete-time set invariance theory. The proposed DCBF provides necessary and sufficient conditions for certifying control invariance and can be used to synthesize a constrained control policy. Moreover, the DCBF can be constructed for arbitrary sets of state and input constraints by taking advantage of maximal output admissible set theory. The resulting DCBF-based controller is proven to be safe and recursively feasible. Numerical examples showcase the effectiveness of the scheme by comparing it to existing constrained control approaches.

Stabilization using shifted equilibria for saturated discrete-time linear systems⋆

We discuss controller designs for asymmetric saturated discrete-time linear systems. Under the assumption that a locally stabilizing controller of the origin is known, we augment the original controller with an additional term that vanishes in a neighborhood of the origin. The augmented controller outperforms the original controller in terms of the estimate of the region of attraction. The paper translates the results discussed in Braun et al. (2022a,b), from the continuous-time setting to the discrete-time setting, and numerically verifies that the results derived for continuous-time systems are recovered if the discrete-time system is obtained through an Euler discretization of a continuous-time system with a sufficiently small sampling rate.

Model Predictive Path Following Control without terminal constraints for holonomic mobile robots

In this paper, a model predictive path following control (MPFC) for holonomic mobile robots is considered. The MPFC is aimed to control a mobile robot to follow a geometric path, where the time evolution of the path parameterization is not fixed a priori. Instead, the time evolution is considered as an extra degree of freedom of the controller. Contrary to previous works, in this study, the closed-loop asymptotic stability of holonomic mobile robots under MPFC without terminal constraints or costs is rigorously proven and a stabilizing-horizon length is calculated. The analysis is based on verifying the cost-controllability assumption by deriving an upper bound of the MPFC value function with a finite prediction horizon. Then, using this bound, the length of a stabilizing prediction horizon is calculated. The analysis is performed in the discrete time settings and theoretical results are verified with numerical simulations as well as implementations on an actual mobile robot.

Stability of Logarithmic Sobolev Inequalities Under a Noncommutative Change of Measure

We generalize Holley-Stroock's perturbation argument from commutative to quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov process can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel.

Asymptotic F test in regressions with observations collected at high frequency over long span

This paper proposes tests of linear hypotheses when the variables may be continuous-time processes with observations collected at a high sampling frequency over a long span. Utilizing series long run variance (LRV) estimation in place of the traditional kernel LRV estimation, we develop easy-to-implement and more accurate F tests in both stationary and nonstationary environments. The nonstationary environment accommodates exogenous regressors that are general semimartingales. Endogenous regressors are allowed in a nonstationary environment similar to cointegration models in the usual discrete-time setting. The F tests can be implemented in exactly the same way as in the discrete-time setting. The F tests are, therefore, robust to the continuous-time or discrete-time nature of the data. Simulations demonstrate the improved size accuracy and competitive power of the F tests relative to existing continuous-time testing procedures and their improved versions. The F tests are of practical interest as recent work by Chang et al. (2021) demonstrates that traditional inference methods can become invalid and produce spurious results when continuous-time processes are observed on finer grids over a long span.

A framework for machine learning of model error in dynamical systems

The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model asmodel error.Our formulation is agnostic to the chosen machine learning model, is presented in both continuous- and discrete-time settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless.First, we study memoryless linear (w.r.t. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root ofTT, the time-interval over which training data is specified.Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent error to recent work on supervised learning between Banach spaces using random features.Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models.

Decomposing reach set computations with low-dimensional sets and high-dimensional matrices (extended version)

Approximating the set of reachable states of a dynamical system is an algorithmic way to rigorously reason about its safety. Despite progress on efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in practice. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach-set computations such that set operations are performed in low dimensions, while matrix operations are performed in the full dimension. Our method is applicable in both dense- and discrete-time settings. For a set of standard benchmarks, we show a speed-up of up to two orders of magnitude compared to the respective state-of-the-art tools, with only modest loss in accuracy. For the dense-time case, we show an experiment with more than 10,000 variables, roughly two orders of magnitude higher than possible before.

Stability analysis and stabilization of linear symmetric matrix-valued continuous, discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.

Starting small in project choice: A discrete-time setting with a continuum of types

We add to the literature on long-term relationships with variable stakes and incomplete information by analyzing a discrete-time trust game between a principal and agent, with a continuum of types. In each period the principal selects the level of a project and the agent then decides whether to cooperate or betray; payoffs in the period scale with the level. The agent's benefit of betraying is privately known. The discrete-time framework allows for analysis of renegotiation in terms of an internal consistency condition that compares actual equilibria in the continuation of the game from any period, improving on the prior literature. Our condition assumes the principal has full power to alter the equilibrium selection. Our main result shows that the resulting perfect Bayesian equilibria converge as the period length shrinks to zero, and we provide a closed-form solution. In equilibrium, the relationship starts small and the level gradually rises until it reaches its maximum; cooperation is viable regardless of the type distribution.

Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

Stability contractors, based on interval analysis, were introduced in recent work as a tool to verify stability domains for nonlinear dynamic systems. These contractors rely on the property that - in case of provable asymptotic stability - a certain domain in a multi-dimensional state space is mapped into its interior after a certain integration time for continuous-time processes or after a certain number of discretization steps in a discrete-time setting. However, a disadvantage of the use of axis-aligned interval boxes in such computations is the omnipresent wrapping effect. As shown in this contribution, the replacement of classical interval representations by ellipsoidal domain enclosures reduces this undesirable effect. It also helps to find suitable ratios for the edge lengths if interval-based domain representations are investigated. Moreover, ellipsoidal domains naturally represent the possible regions of attraction of asymptotically stable equilibrium points that can be analyzed with the help of quadratic Lyapunov functions, for which stability criteria can be cast into linear matrix inequality (LMI) constraints. For that reason, this paper further presents possible interfaces of ellipsoidal enclosure techniques with LMI approaches. This combination aims at the maximization of those domains that can be proven to be stable for a discrete-time range-only localization algorithm in robotics. There, an Extended Kalman Filter (EKF) is applied to a system for which the dynamics are characterized by a discrete-time integrator disturbance model with additive Gaussian noise. In this scenario, the measurement equations correspond to the distances between the object to be localized and beacons with known positions.

Complete Entropic Inequalities for Quantum Markov Chains

We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or L_2-estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.

Backstepping control for a class of underactuated nonlinear mechanical systems with a novel coordinate transformation in the discrete-time setting

In this technical note, a combined discrete-time controller, consisting of a partial feedback linearization controller and a backstepping controller, is proposed to manage the stabilization for a class of underactuated nonlinear mechanical systems. The traditional backstepping control method is impracticable to directly apply it to underactuated systems since they are generally not in the lower triangular form. First, a feedback controller that partially linearizes the underactuated mechanical systems is derived to facilitate the backstepping controller design. In the design, discrete-time system dynamics obtained utilizing the Euler approximation are considered. Afterward, a novel coordinate transformation is introduced for a class of underactuated systems that transform the mathematical model of the partially linearized underactuated mechanical systems into the strict-feedback form. After these steps, the conventional backstepping approach is structured in discrete-time setting. The stability of the closed-loop system dynamics with the proposed combined discrete-time controller is shown utilizing Lyapunov theory. Several computer-based simulations are executed to depict the performance and the applicability of the proposed method on the Cart–Pole system and the Inertia Wheel Pendulum. Moreover, the effectiveness of the proposed nonlinear controller is verified by numerical simulation results in comparison with a discrete-time chattering-free sliding-mode controller.

Kinetic Yvon-Born-Green theory of the linear dielectric constant and complex permittivity of isotropic polar fluids

The theory of the linear static dielectric constant and linear complex permittivity of isotropic polar fluids is formulated starting from the coupled Langevin equationsdescribing the rototranslational dynamics of long-range interacting molecules with thermal agitation and subjected to external forces and torques. To this aim, adequate reduced densities are introduced and equationsgoverning their dynamics derived. In the equilibrium zero frequency limit, integral expressions for the Kirkwood correlation factor g_{K} are given, transparently showing that the popular method consisting in comparing g_{K} with 1 in order to deduce pair dipolar ordering has no serious theoretical grounding. In the dynamical situation, the complex permittivity spectrum of a simple liquid is shown to exhibit an infinite discrete set of relaxation times, some of which may have thermally activated behavior. The theory is also shown to contain all previous results derived in the area provided molecular inertial effects are ignored, so restricting the range of validity of the theory to frequencies much below the far-infrared region. Finally, the theory can be adapted without much effort to relaxation of interacting magnetic nanoparticles for which macroscopic magnetic anisotropy arising from the assembly of nanoparticles is neglected.

On the Oval Shapes of Beach Stones

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

Optimal Asset Allocation Subject to Withdrawal Risk and Solvency Constraints

This paper investigates the optimal asset allocation of a financial institution whose customers are free to withdraw their capital-guaranteed financial contracts at any time. In accounting for the asset-liability mismatch risk of the institution, we present a general utility optimization problem in a discrete-time setting and provide a dynamic programming principle for the optimal investment strategies. Furthermore, we consider an explicit context, including liquidity risk, interest rate, and credit intensity fluctuations, and show by numerical results that the optimal strategy improves both the solvency and asset returns of the institution compared to a standard institutional investor’s asset allocation.