Asymptotic Expansion Research Articles

The maximum norm error estimate and Richardson extrapolation methods of a second-order box scheme for a hyperbolic-difference equation with shifts

The study aims at the development and theoretical analyses of a box method used to solve a kind of hyperbolic equations with shifts. By using the discrete energy method, it is shown that numerical solutions converge to exact solutions with an order of O(τ 2 + h 2) in L ∞-norm as τ = O(h). Here, τ and h are temporal and spatial meshsizes, respectively. According to local truncation error, the asymptotic expansion formula of the numerical solutions is derived by introducing two auxiliary problems. By applying this asymptotic expansion formula, a class of Richardson extrapolations methods have been derived to achieve extrapolation solutions with an order of O(τ 4 + h 4) in L ∞-norm. Numerical results show the high performance of the proposed methods and the exactness of theoretical findings.

Human-in-the-Loop Consensus Control for Multiagent Systems With External Disturbances.

In this article, the human-in-the-loop leader-follower consensus control problem is addressed for multiagent systems (MASs) with unknown external disturbances. A human operator is deployed to monitor the MASs' team by transmitting an execution signal to a nonautonomous leader in response to any hazard detected, with the control input of the leader unknown to all followers. For each follower, a full-order observer, in which the observer error dynamic system decouples the unknown disturbance input, is designed for asymptotic state estimation. Then, an interval observer is constructed for the consensus error dynamic system, where the unknown disturbances and control inputs of its neighbors and its disturbance are treated as unknown inputs (UIs). To process the UIs, a new asymptotic algebraic UI reconstruction (UIR) scheme is proposed based on the interval observer, and one of the significant features of the UIR is the capacity to decouple the control input of the follower. The subsequent human-in-the-loop asymptotic convergence consensus protocol is developed by applying an observer-based distributed control strategy. Finally, the proposed control scheme is validated through two simulation examples.

Analysis of the asymptotic convergence of periodic solution of the Mackey–Glass equation to the solution of the limit relay equation

Analysis of the asymptotic convergence of periodic solution of the Mackey–Glass equation to the solution of the limit relay equation

General backreaction force of cosmological bubble expansion

The gravitational-wave energy-density spectra from cosmological first-order phase transitions crucially depend on the terminal wall velocity of asymptotic bubble expansion when the driving force from the effective potential difference is gradually balanced by the backreaction force from the thermal plasma. Much attention has previously focused on the backreaction force acting on the bubble wall alone but overlooked the backreaction forces on the sound shell and shock-wave front, if any, which have been both numerically and analytically accomplished in our previous studies but only for a bag equation of state. In this paper, we will generalize the backreaction force on bubble expansion beyond the simple bag model. Published by the American Physical Society 2024

Consensus dynamics under asymmetric interactions in low dimensions

AbstractWe consider a set $$N = \{ 1,\ldots ,n \}$$ N = { 1 , … , n } , $$n\ge 2$$ n ≥ 2 , of interacting agents whose individual opinions $$ x_{i}$$ x i , with $$i \in N $$ i ∈ N , take values in some domain $$\mathbb {D}\subseteq \mathbb {R}$$ D ⊆ R . The interaction among the agents represents the degree of reciprocal influence which the agents exert upon each other and it is expressed by a general asymmetric interaction matrix with null diagonal and off-diagonal coefficients in the open unit interval. The present paper examines the asymmetric generalization of the linear consensus dynamics model discussed in previous publications by the same authors, in which symmetric interaction was assumed. We are mainly interested in determining the form of the asymptotic convergence towards the consensual opinion. In this respect, we present some general results plus the study of three particular versions of the linear consensus dynamics, depending on the relation between the interaction structure and the degrees of proneness to evaluation review of the various individual opinions. In the general asymmetric case, the analytic form of the asymptotic consensual solution $$\tilde{x}$$ x ~ is highly more complex than that under symmetric interaction, and we have obtained it only in two low-dimensional cases. Nonetheless, we are able to write those complex analytic forms arranging the numerous terms in an intelligible way which might provide useful clues to the open quest for the analytic form of the asymptotic consensual solution in higher-dimensional cases.

Unorientable topological gravity and orthogonal random matrix universality

The duality of Jackiw-Teitelboim (JT) gravity and a double scaled matrix integral has led to studies of the canonical spectral form factor (SFF) in the so called τ−scaled limit of large times, t → ∞, and fixed temperature, in order to demonstrate agreement with universal random matrix theory (RMT). Though this has been established for the unitary case, extensions to other symmetry classes requires the inclusion of unorientable manifolds in the sum over geometries, necessary to address time reversal invariance, and regularization of the corresponding prime geometrical objects, the Weil-Petersson (WP) volumes. We report here how universal signatures of quantum chaos, witnessed by the fidelity to the Gaussian orthogonal ensemble, emerge for the low-energy limit of unorientable JT gravity, i.e. the unorientable Airy model/topological gravity. To this end, we implement the loop equations for the corresponding dual (double-scaled) matrix model and find the generic form of the unorientable Airy WP volumes, supported by calculations using unorientable Kontsevich graphs. In an apparent violation of the gravity/chaos duality, the τ−scaled SFF on the gravity side acquires both logarithmic and power law contributions in t, not manifestly present on the RMT side. We show the expressions can be made to agree by means of bootstrapping-like relations hidden in the asymptotic expansions of generalized hypergeometric functions. Thus, we are able to establish strong evidence of the quantum chaotic nature of unorientable topological gravity.

ЭКИНЧИ ТАРТИПТЕГИ ТУУНДУСУНА КАРАТА ЧЕЧИЛБЕГЕН ДИФФЕРЕНЦИАЛДЫК ТЕҢДЕМЕ ҮЧҮН ЧЕКТИК МАСЕЛЕНИН ЧЫГАРЫЛЫШЫН КОЛЛОКАЦИЯ-АСИМПТОТИКАЛЫК МЕТОД МЕНЕН ТАБУУ

Boundary value problems for ordinary differential equations is one of the sections of the general theory of differential equations. The difference between the boundary value problem and the Cauchy problem is that the solution of the differential equation must satisfy boundary conditions connecting the values of the desired function at more than one point. The most common boundary value problem refers to two-point boundary value problems for which boundary conditions are set at two points at the ends of the interval on which solutions are being sought. In addition, the special case of two-point boundary value problems also covers such an important section of the theory of differential equations as the theory of oscillations. In this article, the problem of constructing an asymptotic expansion in a small parameter for solving a boundary value problem for a second-order differential equation of an unresolved relative to the highest derivative of the desired solution is considered. The solution of the boundary value problem is found according to the algorithm of the collocation-asymptotic method. A numerical example of a boundary value problem is considered and an asymptotic decomposition is constructed for solving problems with a relatively small parameter, including the first two terms of the expansions.

Study of the direct 16O(p,γ)17F astrophysical capture reaction within a potential model approach

A potential model is applied for the analysis of the astrophysical direct nuclear capture process 16O(p,γ)17F. The phase-equivalent potentials of the Woods-Saxon form for the p−16O interaction are examined which reproduce the binding energies and the empirical values of ANC for the 17F(5/2+) ground and 17F(1/2+) (E⁎=0.495 MeV) excited bound states from different sources. The best description of the experimental data for the astrophysical S factor is obtained within the potential model which yields the ANC values of 1.043 fm−1/2 and 75.484 fm−1/2 for the 17F(5/2+) ground and 17F(1/2+) excited bound states, respectively. The zero-energy astrophysical factor S(0)=9.321 KeV b is obtained by using the asymptotic expansion method of D. Baye. The calculated reaction rates within the region up to 1010 K are in good agreement with those from the R-matrix approach and the hierarchical Bayesian model in both absolute values and temperature dependence.

Vertical line time domain green function and its applications in numerical simulation of ship seakeeping performance

In present study, a vertical line source time domain Green function is proposed and the subsequent numerical evaluation methods are described in detail. For which, Taylor expansion and Chebyshev approximation algorithms are originally extended from point source to line source for small parameters, as well as asymptotic expansions are extended for large parameters. Meanwhile, polynomials are originally proposed to replace the asymptotic expansions to improve the computational efficiency. The accuracy and stability of the proposed method are validated through several cases and good agreement can be obtained.Furthermore, a hybrid Green function method based on the vertical line source Green function and Rankine source is developed for time domain simulation of seakeeping performances of ships. With respect to the hybrid method, the flow domain is divided into inner part and outer part by an artificial control surface. The first-order Taylor Expansion Boundary Element Method (TEBEM) based on the Rankine panel distribution is applied in the inner part, while three-dimensional panel method based on the vertical line source Green function distribution is adopted in the outer part. Under this circumstance, motions of the single-ship (Slender Wigley and Blunt Wigley) and two-ship (two side-by-side ships) in head waves are numerically investigated. The numerical results agree well with the experimental data, which proved the feasibility and the accuracy of present hybrid Green function method for motion prediction of both single-ship and two-ship.

Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions

We study the small-time global approximate controllability for incompressible magnetohydrodynamic (MHD) flows in smoothly bounded two- or three-dimensional domains. The controls act on arbitrary nonempty open portions of each connected boundary component, while linearly coupled Navier slip-with-friction conditions are imposed along the uncontrolled parts of the boundary. Some choices for the friction coefficients give rise to interacting velocity and magnetic field boundary layers. We obtain sufficient dissipation properties of these layers by a detailed analysis of the corresponding asymptotic expansions. For certain friction coefficients, or if the obtained controls are not compatible with the induction equation, an additional pressure-like term appears. We show that such a term does not exist for problems defined in planar simply-connected domains and various choices of Navier slip-with-friction boundary conditions.

Nearly optimal fixed time sliding mode controller for leader–follower consensus problem with partially unknown nonlinear agents

This paper presents a conceptual framework for addressing the consensus problem in multi-agent systems with unknown nonlinear dynamics using reinforcement learning (RL) based nearly optimal sliding mode controller (SMC). The agents’ dynamics are assumed to have uncertainty and mismatched disturbance. An adaptive fixed-time estimator is introduced to estimate uncertain dynamics and disturbances for each agent at a certain time. The paper proposes two control strategies. In the first strategy, a controller is designed, incorporating adaptive SMC and an optimal controller. Adaption law in SMC estimates the bound of fixed time estimator error before convergence, ultimately achieving asymptotic convergence to the sliding surface and converting the agent’s dynamics to linear. This enables solving a linear consensus problem using an RL-based adaptive optimal controller through an on-policy critic–actor method. The second control strategy enhances the adaptive SMC into a fixed-time controller, reducing the time to reach the sliding surface regardless of initial conditions. Consequently, the convergence time of the consensus error to zero is diminished. This reduction in reaching time results in faster convergence of the consensus error to zero. The effectiveness of both strategies is validated through numerical experiments on two real system models, aligning with the theoretical proofs.

Robust reference governor for input-constrained model predictive control to enforce state constraints at low computational cost

In the setting of discrete-time linear systems with unmeasured set-bounded disturbances, this paper proposes a control scheme that combines a disturbance free input-constrained Model Predictive Control (uMPC) policy and a reference governor (RG) which modifies the reference command to enforce constraints that are not enforced by the uMPC policy. Such a scheme, referred to as the robust RGMPC, can handle (possibly nonlinear) state and input constraints and only requires solving an optimisation problem for MPC with polytopic input constraints, for which fast algorithms exist. Conditions are given which ensure recursive feasibility of the robust RGMPC scheme, finite-time convergence of the modified reference command to the desired reference command and asymptotic convergence of the state to an invariant neighbourhood of the associated set-point. Simulation results for a constrained spacecraft rendezvous manoeuvre demonstrate that the rRGMPC scheme has lower average computational time than several robust state and input-constrained MPC controllers.

Fully developed flow through shrouded-fin arrays: exact and asymptotic solutions

The flow resistance, i.e. friction factor times Reynolds number ( $\,f\,{Re}$ ), of longitudinal-fin heat sinks with or without clearance between a shroud and the tips of the fins is an important parameter in thermal design. This is because it dictates the caloric resistance of the heat sink, i.e. change in bulk temperature of the fluid flowing through it. When there is no clearance and the common and oft-valid assumption of negligible fin thickness is invoked, $f\,{Re}$ corresponds to simply that of a rectangular duct. However, with clearance, only numerical results are available as per the well-known study by Sparrow, Baliga and Patankar (ASME J. Heat Transfer, vol. 100, 1978). We develop analytical formulae for $f\,{Re}$ for fully developed flow with clearance. The exact solution is provided by an integral formula derived via conformal mappings. Additionally, simple formulae are derived via asymptotic expansions in three cases: (1) the fin spacing is small compared to the fin height and clearance; (2) the clearance is small compared to the fin spacing, which is small compared to the fin height; (3) the same as case (2) but valid for larger clearances. The different asymptotic formulae are compared to the exact formula, and together cover almost the entire relevant parameter range (for fin spacing and clearance) with errors of less than 15 %. A formula for the limiting case of no clearance is shown to be more accurate, for any fin spacing, than a widely used correlation from the literature.

Lubrication flow in grinding

In the machining process known as grinding, fluid is applied to regulate the temperature of the workpiece and reduce the risk of expensive thermal damage. The factors that influence the transport of this grinding fluid are not well understood; however, it is important to gain understanding in order to try to avoid the unnecessary cost incurred from its inefficient application. In this work, we use the method of matched asymptotic expansions to derive the multiscale system of equations that governs the flow. Under the lubrication approximation, we show that it is possible to calculate the flow rate through the grinding zone without having to solve for the flow far from the grinding zone. Additional empirically determined boundary conditions do not need to be imposed. With this lubrication model, we quantify the effect of experimental parameters on the flow field in the grinding zone and study how the flow regime responds to changes in these parameters.

Large-N integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM: when resurgence meets modularity

Exact expressions for certain integrated correlators of four half-BPS operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory with gauge group SU(N) have been recently obtained thanks to a beautiful interplay between supersymmetric localisation and modular invariance. The large-N expansion at fixed Yang-Mills coupling of such integrated correlators produces an asymptotic series of perturbative terms, holographically related to higher derivative interactions in the low energy expansion of the type IIB effective action, as well as exponentially suppressed corrections at large N, interpreted as contributions from coincident (p, q)-string world-sheet instantons. In this work we define a manifestly modular invariant Borel resummation of the perturbative large-N expansion of these integrated correlators, from which we extract the exact non-perturbative large-N sectors via resurgence analysis. Furthermore, we show that in the ’t Hooft limit such modular invariant non-perturbative completions reduce to known resurgent genus expansions. Finally, we clarify how the same non-perturbative data is encoded in the decomposition of the integrated correlators based on SL(2, ℤ) spectral theory.

Fatigue cracking prediction of RIOHTrack full-scale test track based on variable-order fractional Burgers model

We propose a modified thermal cracking (MTC) model in this paper, which utilizes the evolution of crack propagation to predict the number and density of cracks that will form on the RIOHTrack full-scale test track. We incorporate the variable-order fractional Burgers model as an important component of the thermal cracking (TC) model, for calculating the creep compliance. Furthermore, we apply the Levenberg–Marquardt (LM) algorithm as a data-driven approach, for parameter fitting. The MTC model is constructed technically through creep compliance approximation compute and parameter fitting as follows. First, we employ the variable-order fractional Burgers model to calculate the asymptotic creep compliance of each pavement segment within the RIOHTrack full-scale test track. More precisely, we derive the explicit asymptotic expansion of creep compliance based on the variable-order fractional Burgers model. Such an asymptotic expansion represented by the Gamma function is a simplified representation of the creep compliance, and it ensures that the fitting accuracy R2 is greater than 0.87. Parameter fitting of the creep compliance is based on the RIOHTrack full-scale pavement test data, using the LM algorithm. Then, the creep compliance in the original TC model is mathematically upgraded to the above variable-order fractional approximation form, and parameters in the MTC model are numerically estimated using the LM algorithm based on prior knowledge from the RIOHTrack test data. Our prediction analyses based on the RIOHTrack data demonstrate the superior performance of the proposed MTC model.

Polynomial Optimization Over Unions of Sets

AbstractThis paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The application for computing (p, q)-norms of matrices is also presented.

Scaling solutions for current-carrying cosmic string networks

Cosmic string networks are the best motivated relics of cosmological phase transitions, being unavoidable in many physically plausible extensions of the Standard Model. Most studies, including those providing constraints from and forecasts of their observational signals, rely on assumptions of featureless networks, neglecting the additional degrees of freedom on the string worldsheet, e.g., charges and currents, which are all but unavoidable in physically realistic models. An extension of the canonical velocity-dependent one-scale model, accounting for all such possible degrees of freedom, has been recently developed. Here we improve its physical interpretation by studying and classifying its possible asymptotic scaling solutions, and in particular how they are affected by the expansion of the Universe and the available energy loss or transfer mechanisms. We find three classes of solutions. For sufficiently fast expansion rates the charges and currents decay and one asymptotes to the Nambu-Goto case, while for slower expansion rates they can dominate the network dynamics. In between the two there is a third regime in which the network, including its charge and current, reaches full scaling. Under specific but plausible assumptions, this intermediate regime corresponds to the matter-dominated era. Our results agree with, and significantly extend, those of previous studies. Published by the American Physical Society 2024

Asymptotics of Saran's hypergeometric function FK

In this paper, we first establish asymptotic expansions of the Humbert function Ψ1 for one large variable. The resulting expansions are then used to derive an asymptotic expansion of Saran's hypergeometric function FK when two of its variables become simultaneously large.

A Moment-Sum-of-Squares Hierarchy for Robust Polynomial Matrix Inequality Optimization with Sum-of-Squares Convexity

We study a class of polynomial optimization problems with a robust polynomial matrix inequality (PMI) constraint where the uncertainty set itself is also defined by a PMI. These can be viewed as matrix generalizations of semi-infinite polynomial programs because they involve actually infinitely many PMI constraints in general. Under certain sum-of-squares (SOS)-convexity assumptions, we construct a hierarchy of increasingly tight moment-SOS relaxations for solving such problems. Most of the nice features of the moment-SOS hierarchy for the usual polynomial optimization are extended to this more complicated setting. In particular, asymptotic convergence of the hierarchy is guaranteed, and finite convergence can be certified if some flat extension condition holds true. To extract global minimizers, we provide a linear algebra procedure for recovering a finitely atomic matrix-valued measure from truncated matrix-valued moments. As an application, we are able to solve the problem of minimizing the smallest eigenvalue of a polynomial matrix subject to a PMI constraint. If SOS convexity is replaced by convexity, we can still approximate the optimal value as closely as desired by solving a sequence of semidefinite programs and certify global optimality in case that certain flat extension conditions hold true. Finally, an extension to the nonconvexity setting is provided under a rank 1 condition. To obtain the above-mentioned results, techniques from real algebraic geometry, matrix-valued measure theory, and convex optimization are employed. Funding: This work was supported by the National Key Research and Development Program of China [Grant 2023YFA1009401] and the National Natural Science Foundation of China [Grants 11571350 and 12201618].