Singular Problems Research Articles

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

Singular control of (reflected) Brownian motion: a computational method suitable for queueing applications

AbstractMotivated by applications in queueing theory, we consider a class of singular stochastic control problems whose state space is the d-dimensional positive orthant. The original problem is approximated by a drift control problem, to which we apply a recently developed computational method that is feasible for dimensions up to $$d=30$$ d = 30 or more. To show that nearly optimal solutions are obtainable using this method, we present computational results for a variety of examples, including queueing network examples that have appeared previously in the literature.

Correction: Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Correction: Solution of linear and nonlinear singular value problems using operational matrix of integration of Fibonacci wavelets

Existence Results Related to a Singular Fractional Double-Phase Problem in the Whole Space

In this paper, we will study a singular problem involving the fractional (q1(x,.)-q2(x,.))-Laplacian operator in the whole space RN,(N≥2). More precisely, we combine the variational method with monotonicity arguments to prove that the associated functional energy admits a critical point, which is a weak solution for such a problem.

Non-singular control strategies for a class of strict-feedback nonlinear time-delay systems

This paper focuses on the non-singular control for strict-feedback nonlinear time-delay systems. The boundedness of all the signals of the closed-loop system is rigorously proved by introducing a class of Lyapunov-Krasovskii functionals, including exponential functions. Besides, novel auxiliary functions with continuous derivatives are proposed to construct the control law to avoid the discontinuity of the virtual controllers. Besides, the amount of computation is reduced by applying the new auxiliary functions when the other control methods require loads of calculations for handling singular problems. In the end, numerical examples are given to validate the effectiveness of the proposed control scheme.

Debugging Pathways: Open-Ended Discrepancy Noticing, Causal Reasoning, and Intervening

Learning to respond to a computer program that is not working as intended is often characterized as finding a singular bug causing a singular problem. This framing underemphasizes the wide range of ways that students and teachers could notice discrepancies from their intention, propose causes of those discrepancies, and implement interventions. Weaving together a synthesis of the existing research literature with new multimodal interaction analyses of teacher–student conversations during coding, we propose a framework for debugging that foregrounds this open-endedness. We use the framework to structure an analysis of three naturalistic debugging situations (with US 5th–10th graders) that range from solo debugging to collaborative discourse. We argue that a broken computer program is a polysemous object through which teachers and students actively and publicly notice, reason about, and negotiate different debugging pathways. We document students and teachers improvisationally altering a debugging pathway, justifying a particular pathway, and outwardly discussing competing pathways. This paper provides a framework for structuring debugging pedagogy to be less about scaffolding a student toward a specific pathway to a fix and more about exploring multiple possible pathways and judging the (learning) value of various routes.

Singular elliptic measure data problems with irregular obstacles

We investigate elliptic irregular obstacle problems with p-growth involving measure data. Emphasis is on the strongly singular case 1<p≤2−1/n, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.

Governing the global fisheries commons

Despite significant advances in understanding the biophysical and institutional causes of overfishing, we have yet to make progress in addressing the depletion of our global fisheries stock. Investigations of potential solutions tend to be too broad (mischaracterizing global fisheries as a singular commons problem to be addressed at the supranational level) or too narrow (focusing on improving management of small fisheries at the micro level). This article attempts to bridge the gap between our scientific understanding of our collective dilemmas and their pragmatic solutions. Building on insights from Nobel laureate Elinor Ostrom, we frame the depletion of global fisheries as a nested set of diverse and interconnected collective action problems organized at different horizontal and vertical levels, where decisions and actions of one jurisdictional unit reinforce and amplify problems (and solutions) for other units. We examine features of the global fisheries system, such as nonstationarity, nestedness, and prohibitive transaction costs. Then, we explore some potential solutions. The success of our conservation goals depends on our ability to craft institutional rules at the lower levels that are adaptive to local conditions, address incentive misalignment issues, and allow for the transfer of positive externalities to adjacent and higher levels.

Regularizing effect of absorbtion term in singular and degenerate elliptic problems

Dans cet article, nous étudions l’existence et la régularité des solutions au problème singulier suivant\\\begin{equation}\left\{\begin{array}{lll}-\displaystyle\mbox{div} \big(a(x,u)\vert\nabla u\vert^{p-2}\nabla u\big) + \vert u\vert^{s-1}u =h(u)f &amp;\mbox{ in } \Omega \\&amp;u\geq 0 &amp;\mbox{ in }\Omega \\&amp;u=0 &amp;\mbox{ on } \delta\Omega\\\end{array}\right.\end{equation} prouvant que le terme d’ordre inférieur $u\vert u\vert^{s-1}$ a des effets régularisants sur les solutions dans le cas d’un opérateur elliptique à coercivité dégénérée.

Cosmological Consequence of Varying Speed of Light and Gravitational Constant

&lt;p&gt;The speed of light is taken to be a constant in a vacuum. This forms the basic tool for the principle of General Covariance, which asserts that all laws of Physics should take the same form in all frames of reference. Without putting inflation into consideration, the theory of varying speed of light (VSL) would solve basic problems of cosmology in the early universe. Furthermore, the Gravitational constant G that occurred in the Friedmann Equations may not have been real constants in the early universe but have some variation with the universe scale factor. Cosmological models with varying physical constants have been of interest in recent years with few works in the literature. A cosmological solution obtained by incorporating variable speed of light and gravitational constant gives a cosmic model which is free from the initial Big Bang singularity and horizon problem. It is also observed here that the early universe was dominated by dark matter, however, as the scale factor increases, the dark energy become dominant.&lt;/p&gt;

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

From Aerial to underwater: Modeling and trajectory tracking control of the underactuated quadrotor hybrid Aerial underwater vehicle

The underactuated Quadrotor Hybrid Aerial Underwater Vehicle (HAUV) can operate in multiple mediums and seamlessly transition between them. Affected by the buoyancy, underwater control suffers a critical 180° turnover singular problem. To address this challenge, this paper presents a detailed underwater 6-degree of freedom dynamics based on the unit quaternions. A novel geometric attitude extraction method is introduced to improve the classical geometric control. Subsequently, a cascaded controller is developed for trajectory tracking. This controller consists of a position loop and an attitude loop, integrated through the attitude extraction method. It extracts the desired attitude for the inner loop from the desired thrust direction in the outer loop. Rigorous analysis demonstrates that the closed system exhibits global asymptotic stability, effectively mitigating the singular problem. Simulations demonstrate that the controller effectively adapts the quadrotor HAUV, regardless of its buoyancy (positive, negative, or neutral), to accurately track underwater trajectories and fixed points.

Finite-Time Continuous Nonsingular Terminal Modified Adaptive-Gain Super-Twisting Control: Application to a 2-DOF Planar Robot Manipulator System.

This article proposes finite-time continuous nonsingular terminal modified adaptive-gain super-twisting control (FT-CNT-MAG-STC) for the second-order disturbed systems. Compared with existing sliding-mode controllers, the noteworthy improvements of the proposed framework are the fast finite-time convergence, continuous control signal, ease-of-implementation feature, and relaxation of the assumption related to the information on the bounds of the disturbance and its derivative. In the proposed framework, a fast nonsingular terminal sliding surface is first developed such that the singularity problem is avoided and the convergence rate is improved. Then, we design a continuous modified super-twisting algorithm with an adaptive gain, under which the need for the bounds information of the disturbance and its derivative is relaxed and the performance of the closed-loop system is enhanced. Rigorous analysis is provided to prove the finite-time convergence of the system states to small regions containing the origin. We apply the proposed framework to the position control for a 2-DOF planar robot manipulator system. Various experimental results are illustrated to evaluate the effectiveness of the designed position controller.

Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis.

A Novel Faster Fixed-Time Adaptive Control for Robotic Systems With Input Saturation

In this paper, an adaptive anti-saturation fixed-time control method with a faster convergence rate is studied for uncertain robotic systems. Firstly, a new segmental sliding variable is constructed to solve the singularity problem brought by the terminal sliding mode control (TSMC) and achieve a faster convergence rate. Secondly, to approximate and compensate for the model uncertainty and the viscous friction parameter, an adaptive neural network (ANN) is employed. Then, a novel auxiliary system is constructed to mitigate the effects of input saturation. Based on this, a novel non-singular TSMC algorithm integrated with the ANN and the auxiliary system is designed, so that the trajectory tracking errors of the robotic system can converge within a faster fixed time with actuator saturation. Finally, the superiority and practicability of the present method are verified by comparative experiments.

A High-Resolution Exponential Spline Method and Its Convergence Analysis for Two-Point Mixed Boundary Value Problems

In this paper, we present a new numerical method for effectively solving general second-order ordinary differential equations with mixed boundary conditions. Our approach utilizes a quasi-variable mesh enabling us the flexibility to adapt the mesh density according to different boundary layer problems. By employing a discretization technique that incorporates the construction of exponential spline, we achieve third-order accuracy at internal grid points, while boundary points exhibit fourth-order accuracy. Since the method is based on off-step grid points, it eliminates the need to modify the method while employing it to singular problems. We provide computational results to various numerical problems which arise in different physical phenomena like Burger’s equation and Sturm–Liouville equation. A comparison with recent findings underscores the superior performance of our method.

Existence, Multiplicity and $$C^1$$-Regularity for Singular Parametric Problems Driven by the Sum of Three Distinct Anisotropic Operators

Existence, Multiplicity and $$C^1$$-Regularity for Singular Parametric Problems Driven by the Sum of Three Distinct Anisotropic Operators

Existence and regularity results for a class of singular parabolic problems with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} data

In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in L1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\Omega )$$\\end{document}, there exists a solution that “instantly” becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data.

Adaptive neural predefined-time hierarchical sliding mode control of switched under-actuated nonlinear systems subject to bouc-wen hysteresis

For switched under-actuated systems subject to bouc-wen hysteresis, an adaptive neural predefined-time control scheme is presented first time in this paper. To improve the robustness and response rate of the system under consideration, a hierarchical sliding mode control technique is introduced. Based on a predefined-time stability criterion, the stabilization time of the system can be directly preset under the designed controller. Besides, the convergence time depends only on the preset value and is independent of all other parameters. The projection algorithm is introduced to avoid the singularity problem in the controller design process. Meanwhile, it is proved that all signals of the closed-loop system are bounded after a preset time through the Lyapunov stability theory. Finally, a simulation example and a comparison example are given to further illustrate the effectiveness and superiority of developed control approach.

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d &gt; n - 2 2 {d&gt;\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.