First-order System Research Articles

Existence of solutions to contact mean-field games of first order

Abstract This paper deals with the existence of solutions of a class of contact mean-field game systems of first order consisting of a contact Hamilton-Jacobi equation and a continuity equation. Evans found a connection between Hamilton-Jacobi equations and continuity equations from the weak KAM point of view, where the coupling term is zero. Inspired by his work, we prove the main existence result by analyzing the properties of the Mather set for contact Hamiltonian systems.

On shear motions in nonlinear transverse isotropic elastodynamics

We study the propagation of shear waves in an anisotropic incompressible medium composed of an elastic fiber-reinforced material. We first derive the general determining equations of such motions when the fibers are arbitrarily arranged with respect to the direction of propagation of the waves. Then, we examine the “standard reinforced model” so as to root the nonlinearity of the dynamic problem directly to the presence of the reinforcing fibers, and finally, we consider an asymptotic model in terms of the limit of finite but small amplitude of the waves. Clearly, if the fibers are arranged along the direction of propagation of the waves, the mechanical behavior of the material will be isotropic. When the fibers are tilted an amount of the same order of the amplitude of the wave with respect to this direction, the asymptotic first-order reduced system is characterized by a peculiar mixed second-/third-order nonlinearity. This contrasts to what occurs for larger tilting angles where only quadratic nonlinearities must be considered. In the latter case, as we may expect, in the asymptotic limit, we reduce the system to an inviscid classical Burger’s equation. Our results are fundamental to the study of the dynamics of fiber-reinforced materials and equally when we consider dissipative and/or dispersive effects.

The Heterogeneous Multiscale Method for Dispersive Maxwell Systems

In this work, we apply the finite element heterogeneous multiscale method to a class of dispersive first-order time-dependent Maxwell systems. For this purpose, we use an analytic homogenization result, which shows that the effective system contains additional dispersive effects. We provide a careful study of the (time-dependent) micro problems, including $\mathrm{H}^2$ and micro errors estimates. Eventually, we prove a semi-discrete error estimate for the method.

On numerical methods to second-order singular initial value problems with additive white noise

In this work, we investigate the strong convergence of the Euler–Maruyama method for second-order stochastic singular initial value problems with additive white noise. The singularity at the origin brings a big challenge that the classical framework for stochastic differential equations and numerical schemes cannot work. By converting the problem to a first-order stochastic singular differential system, the existence and uniqueness of the exact solution is studied. Moreover, under some suitable assumptions, it is proved that the Euler–Maruyama method is of (1/2−ɛ) order convergence in mean-square sense, where ɛ is an arbitrarily small positive number, which is different from the consensus that the Euler–Maruyama method is convergent with first order in strong sense when solving stochastic differential equations with additive white noise. While, it is found that if the diffusion coefficient vanishes at the origin, the convergence order in mean-square sense will be increased to 1−ɛ. Our theoretical findings are well verified by numerical examples.

Improved state augmentation method for buffeting analysis of structures subjected to non-stationary wind

Extreme winds such as hurricanes and thunderstorms often present non-stationary characteristics, having time-varying mean wind speeds and non-stationary wind fluctuations. When concerning the wind-induced vibrations under non-stationary wind, the excitation will be a non-stationary process, and the wind-structure coupled system can be represented by a linear time-varying (LTV) system. The aim of this study is to present a state augmentation method to investigate the non-stationary buffeting of a model bridge tower subjected to non-stationary wind with consideration of the aeroelastic damping. Based on the theory of stochastic differential equations and Itô’s lemma, the statistical moments of the non-stationary buffeting response are derived through solving a first-order ordinary differential equation system. The proposed method is validated by comparisons with the Monte Carlo method and the pseudo excitation method. The result shows that the state augmentation method has higher accuracy and efficiency than the well-accepted time–frequency techniques.

Towards a Taylor-Carleman bilinearization approach for the design of nonlinear state-feedback controllers

The Carleman bilinearization is an approach that performs an exact conversion of a finite-dimensional nonlinear system into an infinite-dimensional bilinear system. A finite-dimensional system is later obtained through a truncation for analysis and control purposes. This paper investigates the linear matrix inequality (LMI)-based design of a switched state-feedback control law for the model obtained via Carleman bilinearization of a first-order nonlinear system. It is shown that in order to obtain feasible design conditions, the performance requirements must be relaxed in a neighborhood of the zero equilibrium point, so that problems arising from the uncontrollability of the linear part of the model can be avoided. The effectiveness of the proposed approach is shown using a numerical example and experimental results using a multi-input tank system.

Studies on System Identification of Multiple Input Multiple Output (MIMO) Water Tank

In this study, the system identification of a multiple input multiple-output (MIMO) water tank has been investigated. The given experimental system is a first-order system. Its transfer function was evaluated and the time constant was predicted. Once the parameters of the transfer functions (delay, gain, and time constant) were determined for each step input. This model each of the step responses using MatLab’s Simulink. These simulated responses were then compared to the observed experimental responses.

Evolution equations dynamical system of the Lemaître–Tolman–Bondi metric containing coupled dark energy

In this paper, we consider inhomogeneous spherically symmetric models based on the Lemaître–Tolman–Bondi (LTB) metric, assuming as its source an interactive mixture of ordinary baryonic matter, cold dark matter and dark energy with a coupling term proportional to the addition of energy densities of both dark fluids. We reduce Einstein’s field equations to a first-order seven-dimensional autonomous dynamical system of evolution equations and algebraic constraints. We study in detail the evolution of the energy density and spatial curvature profiles along the phase–space by means of two subspace projections: a three-dimensional projection associated with the solutions of the Friedman–Lemaître–Robertson–Walker metric (invariant subspace) and a four-dimensional projection describing the evolution of the inhomogeneous fluctuations. We also classify and study the critical points of the system in comparison with previous work on similar sources, as well as solving numerically the equations for initial energy density and curvature profiles that lead to a spherical bounce whose collapsing time we estimate appropriately.

Bipartite containment control of multi-agent systems with multiple leader groups

This article mainly considers the bipartite containment control problem of multi-agent systems with first- and second-order dynamics, where the antagonistic networks and multiple leader groups are considered. Specially, under the antagonistic networks, the multiple leader groups constitute two opposite convex hulls. One is a real convex hull composed of the real leader groups and the other is a virtual convex hull composed of opposite virtual leaders. For the first-order multi-agent systems with multiple leader groups, the followers move into the stationary real or virtual convex hull. While for the second-order multi-agent systems, the followers finally move into the dynamic real or virtual convex hull. In both cases, we obtain the sufficient conditions on achieving bipartite containment control. In the end, the effectiveness is verified by some simulations.

Imaging Anisotropic Conductivities from Current Densities

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discretized by the standard Galerkin finite element method. We establish the continuity and differentiability of the forward map with respect to the conductivity tensor in the $L^p(\Omega)^{d,d}$-norms, the existence of minimizers and optimality systems of the regularized formulation using the concept of H-convergence. Further, we provide a detailed analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of H-convergence. In addition, we develop a projected Newton algorithm for solving the first-order optimality system. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.

Green’s Functions for First-Order Systems of Ordinary Differential Equations without the Unique Continuation Property

This paper is a contribution to the spectral theory associated with the differential equation $Ju'+qu=wf$ on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Under these hypotheses it may not be possible to uniquely continue a solution from one point to another, thus blunting the standard tools of spectral theory. Despite this fact we are able to describe symmetric restrictions of the maximal relation associated with $Ju'+qu=wf$ and show the existence of Green's functions for self-adjoint relations even if unique continuation of solutions fails.

The mean-field approximation for higher-dimensional Coulomb flows in the scaling-critical L ∞ space

In the mean-field scaling regime, a first-order system of particles with binary interactions naturally gives rise to a scalar partial differential equation (PDE), which, depending on the nature of the interaction, corresponds to either the Hamiltonian or gradient flow of the effective energy of the system for a large number of particles. The empirical measure of such systems is a weak solution to this limiting mean-field PDE, and one expects that as the number of particles tends to infinity, it converges along its lifespan in the weak-* sense to a more regular solution of the PDE, provided it does so initially. Much effort has been invested over the years in proving and quantifying this convergence under varying regularity assumptions. When the interaction potential is Coulomb, the mean-field PDE has a scaling invariance which leaves the L ∞ norm unchanged; i.e., L ∞ is a critical function space for the equation. Moreover, the L ∞ norm is either conserved or decreasing, and the equation is globally well-posed in this space, making it a natural choice for studying the dynamics. Building on our previous work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431) for point vortices (i.e. d = 2), we prove quantitative convergence of the empirical measure to the L ∞ solution of the mean-field PDE for short times in dimensions d ⩾ 3. To the best of our knowledge, this is the first such work outside of the 2D case. Our proof is based on a combination of the modulated-energy method of Serfaty (2020 Duke Math. J. 169 2887–935) and a novel mollification argument first introduced by the author in Rosenzweig (2022 Arch. Ration. Mech. Anal. 243 1361–431). Compared to our prior work (Rosenzweig 2022 Arch. Ration. Mech. Anal. 243 1361–431), the new challenge is the non-logarithmic nature of the potential.

Existence and uniqueness of solutions of nonlinear fractional stochastic differential systems with nonlocal functional boundary conditions

The authors study the existence and uniqueness of solutions to nonlinear first-order fractional stochastic differential systems driven by Brownian motion and with nonlocal functional boundary conditions. The technique of proof involves Perov’s fixed point theorem with matrices that converge to zero and the Leray–Schauder theorem.

Total Lagrange implementation of a finite-deformation continuum dislocation dynamics model of mesoscale plasticity

We present a computational algorithm for solving the recently developed finite-deformation continuum dislocation dynamics theory of mesoscale plastic deformation of single crystals (Starkey et al., 2020). This CDD theory is based on a vector density representation of dislocations governed by curl-type transport-reaction equations subjected to the divergence-free constraint of the appropriate dislocation density. These density evolution equations are to be solved simultaneously with the finite-deformation crystal mechanics. Specifically, our algorithm aims to solve the referential form of the governing equations for a representative volume element (RVE) subject to remote uniform loading. The mechanical fields at the mesoscale are thus split into RVE-averages plus fluctuating components and treated using a strain-driven homogenization scheme. A virtual work-based total Lagrange formulation was used to discretize the governing mechanics equations. A first-order system least squares finite element formulation was used to solve the transport equations. The two schemes are coupled in a staggered fashion. As a part of the crystal mechanics discretization, we derive a consistent tangent modulus and show that the stress update for this model is both linear and global. This linear stress update comes at the cost of solving the dislocation transport equations at every time step to update the plastic distortion caused by dislocation motion. Several test problems are given, demonstrating the ability of the discretization scheme to solve the problem, including the expansion of dislocation loop-like bundles under constant velocity and driven by a mean deformation gradient, dynamic recovery of two oppositely oriented tilt boundaries in a single crystal, and a uniaxial tension test of a single crystal with one slip system activated. In most of these examples, the evolution behavior of the dislocations in the finite deformation regime is demonstrated.

A Family of A -Stable Optimized Hybrid Block Methods for Integrating Stiff Differential Systems

In this article, a family of one-step hybrid block methods having two intrastep points is developed for solving first-order initial value stiff differential systems that occur frequently in science and engineering. In each method of the family, an intrastep point controls the order of the main method and a second one has a control over the stability features of the method. The approach used to develop the class of A -stable methods is based on interpolation and collocation procedures. The methods exhibit hybrid nature and produce numerical solutions at several points simultaneously. These methods can also be formulated as Runge-Kutta (RK) methods. Comparisons between the RK and block formulations of the proposed methods reveal a better performance of the block formulation in terms of computational efficiency. Furthermore, the efficiency of the methods is improved when they are formulated as adaptive step-size solvers using an error-control approach. Some methods of the proposed class have been tested to solve some well-known stiff differential systems. The numerical experiments show that the proposed family of methods performs well in comparison with some of the existing methods in the scientific literature.

Reinforcement learning-based optimised control for a class of second-order nonlinear dynamic systems

This paper presents an optimised tracking control scheme based on reinforcement learning (RL) for a class of second-order nonlinear systems with unknown dynamics. Different from the first-order dynamic system control, the second-order case is required to synchronously steer two variables of position and velocity, hence it makes this optimised control more challenging to accomplish. To achieve the optimised control, first, neural network (NN) is employed to approximate the solution of Hamilton–Jacobi–Bellman (HJB), and then an RL is performed by constructing both critic and actor based on the NN approximation. Since the RL training laws are derived from the negative gradient of a simple positive function generated in accordance with the partial derivative of HJB equation, it can make the control algorithm significantly simple to compare with the existing optimal control methods. Meanwhile, it can also release the condition of persistence excitation and compensate for the nonlinear uncertainty. Finally, the proposed adaptive control method can guarantee the desired results that are demonstrated by theorem, proof and simulation.

A staggered cell-centered DG method for the biharmonic Steklov problem on polygonal meshes: A priori and a posteriori analysis

In this paper, a staggered cell-centered discontinuous Galerkin method is developed for the biharmonic problem with the Steklov boundary condition. Our approach utilizes a first-order system form of the biharmonic problem and can handle fairly general meshes possibly including hanging nodes, which favors adaptive mesh refinement. Optimal order error estimates in L2 norm can be proved for all the variables. Moreover, the approximation of the primal variable superconverges in L2 norm to a suitably chosen projection without requiring additional regularity. Residual type error estimators are proposed, which can guide adaptive mesh refinement to deliver optimal convergence rates even for solutions with singularity. Numerical experiments confirm that the optimal convergence rates in L2 norm can be achieved for all the variables. Moreover, all the provided residual type error estimators show the desired results. In particular, the numerical results demonstrate that the proposed scheme on a polygonal approximation of the disk works well for the classic Babuška example.

Optimal control for a coupled spin-polarized current and magnetization system

This paper is devoted to an optimal control problem of a coupled spin drift-diffusion Landau–Lifshitz–Gilbert system describing the interplay of magnetization and spin accumulation in magnetic-nonmagnetic multilayer structures, where the control is given by the electric current density. A variational approach is used to prove the existence of an optimal control. The first-order necessary optimality system for the optimal solution is derived in one space-dimension via Lagrange multiplier method. Numerical examples are reported to validate the theoretical findings.

Exponential Convergence Towards Consensus for Non-Symmetric Linear First-Order Systems in Finite and Infinite Dimensions

We consider finite and infinite-dimensional first-order consensus systems with timeconstant interaction coefficients. For symmetric coefficients, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function. We investigate here the convergence to consensus in the non-symmetric case: we identify a positive weight which allows to define a weighted mean corresponding to the consensus, and obtain exponential convergence towards consensus. Moreover, we compute the sharp exponential decay rate.

A new perspective on teaching the natural exponential to engineering students

The natural exponential and logarithm are typically introduced to undergraduate engineering students in a calculus course using the notion of limits. We here present an approach to introduce the natural exponential/logarithm through a novel interpretation of derivatives. This approach does not rely on limits, allowing an early and intuitive introduction of these functions. The question behind our contribution is whether one can introduce derivatives using only polynomials and power series? Motivated by an earlier exposure of engineering students to differential equations, we demonstrate that the natural exponential/logarithm can arise from two common differential equations. Our limit-free approach to derivatives provides an intuitive interpretation of , the Euler number, and an intuitive introduction of time constants in first-order dynamical systems.