Second-order Differential Systems Research Articles

Solvability and optimal control for second-order stochastic differential systems under the influence of delay and impulses

We explore a new kind of second-order impulsive stochastic differential system with delays. By using the Hausdorff measure of noncompactness and M o ¨ nch fixed point theorem, we investigated the existence of mild solution for the considered systems. Further, we proved the existence of optimal control of the proposed system by using minimizing sequence, Lagrange functional, and Marzur lemma. An example is provided to validate the obtained results.

Canonical Equations of Hamilton with Symmetry and Their Applications

Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schrödinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used.

Upper and lower solutions method for a class of second-order coupled systems

This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.

Multiple homoclinic solutions for nonsmooth second-order differential systems

Multiple homoclinic solutions for nonsmooth second-order differential systems

Existence and Approximate Controllability Results for the Second-Order Abstract Neutral Differential System with Damping

Existence and Approximate Controllability Results for the Second-Order Abstract Neutral Differential System with Damping

New Qualitative Outcomes for Ordinary Differential Systems of Second Order

This paper deals with a nonlinear ordinary differential system of second order. In the paper, qualitative properties of solutions of the system called asymptotic stability (AS), uniform stability (US), boundedness, ultimately boundedness (UB) and integrability of solutions, are investigated by using the second method of Lyapunov. We give four new qualitative results and an example as a numerical application of the results. The results of this article extend and improve some earlier ones in the literature.

An analysis on the optimal control results for second-order Sobolev-type delay differential inclusions of Clarke’s subdifferential type

In this paper, we study the existence and optimal control results for second-order Sobolev-type delay systems with Clarke’s subdifferential type. Initially, the existence of mild solution is established for the proposed second-order delay differential system with the novel ideas of Clarke’s subdifferential. The fixed point theorem of condensing multivalued maps, the strongly continuous cosine family, and the properties of Clarke’s subdifferential are used to establish the existence of mild solution. Moreover, the existence of optimal control pair that is governed by the presented system is verified through Balder’s theorem. Finally, an example is provided to illustrate the main results.

Approximate Controllability Outcomes of Impulsive Second-Order Stochastic Neutral Differential Evolution Systems

In this manuscript, we present a collection of suitable requirements for the approximate controllability outcomes of impulsive second-order stochastic neutral differential evolution systems. We use the ideas from the sine and cosine functions and the fixed point strategy to demonstrate the primary findings. The analysis then moves to second-order nonlocal stochastic neutral differential systems. Eventually, in order to make our topic more useful, we will provide an epistemological implementation.

Scaled consensus of second-order nonlinear multi-agent systems with distributed adaptive control via non-reduced order method

In this article, the leaderless and leader-following scaled consensus (SC) of second-order multi-agent systems (SOMASs) with nonlinear dynamics under distributed adaptive control are investigated by non-reduced order method (NROM). Firstly, the NROM is first applied to transform the variable-replaced SOMASs into a pure second-order differential system. Secondly, a novel Lyapunov function involving the error variables and derivative of the error variables is constructed to directly discuss the second-order differential system, which is completely different from the traditional analysis method. Thirdly, two different coupling gains are designed in the light of the position and velocity of the agent to solve the leader-following SC more flexibly. Finally, two numerical examples are cited to verify the practicability of the theoretical derivation.

Event-based fault-tolerant [formula omitted] synchronization for inertial neural networks via a semi-Markov jump approach

In this paper, the synchronization problem of semi-Markov jump neural networks with inertial terms is studied. The semi-Markov jump (s-MJ) process is used to describe the random jump parameters in the system, so Weibull distribution is used to replace the exponential distribution of Markov jump process for the sojourn-time (ST) of each mode in the system. First, a second-order differential system is transformed into a first-order differential system by an appropriate vector substitution. Then, by applying passive fault-tolerant control techniques, the master and slave systems can be synchronized even when the controller fails. In addition, in order to reduce the waste of bandwidth, an event-triggered strategy is designed to determine whether the data needs to be transmitted. Then, by constructing a suitable Lyapunov function and using the characteristics of the cumulative distribution function, the sufficient conditions are established to ensure the stochastic stability of the closed-loop system and meet the specified H∞ performance when the controller fails. Finally, a numerical example and an image processing example are given to verify the effectiveness of the proposed method.

On Asymptotic Properties of Stochastic Neutral-Type Inertial Neural Networks with Mixed Delays

This article studies the stability problem of a class of stochastic neutral-type inertial delay neural networks. By introducing appropriate variable transformations, the second-order differential system is transformed into a first-order differential system. Using homeomorphism mapping, standard stochastic analyzing technology, the Lyapunov functional method and the properties of a neutral operator, we establish new sufficient criteria for the unique existence and stochastically globally asymptotic stability of equilibrium points. An example is also provided, to show the validity of the established results. From our results, we find that, under appropriate conditions, random disturbances have no significant impact on the existence, stability, and symmetry of network systems.

Fundamental response in the vibration control of buildings subject to seismic excitation with ATMD

The linear quadratic regulator for vibration systems subject to seismic excitations is discussed in his own physical newtonian space as a second-order linear differential system with matrix coefficients. The linear quadratic regulator leads to a fourth-order system and second-order transversality conditions. Those systems are studied with a matrix basis generated by a fundamental matrix solution.

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

Positive solutions for boundary value problems of a class of second-order differential equation system

Abstract This article discusses the existence of positive solutions for the system of second-order ordinary differential equation boundary value problems − u ″ ( t ) = f ( t , u ( t ) , v ( t ) , u ′ ( t ) ) , t ∈ [ 0 , 1 ] , − v ″ ( t ) = g ( t , u ( t ) , v ( t ) , v ′ ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , \left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(t)=f\left(t,u\left(t),v\left(t),u^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ -{v}^{^{\prime\prime} }\left(t)=g\left(t,u\left(t),v\left(t),v^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ u\left(0)=u\left(1)=0,\hspace{1em}v\left(0)=v\left(1)=0,\end{array}\right. where f , g : [ 0 , 1 ] × R + × R + × R → R + f,g:\left[0,1]\times {{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{+}\times {\mathbb{R}}\to {{\mathbb{R}}}^{+} are continuous. Under the related conditions that the nonlinear terms f ( t , x , y , p ) f\left(t,x,y,p) and g ( t , x , y , q ) g\left(t,x,y,q) may be super-linear growth or sub-linear growth on x , y , p x,y,p , and q q , we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition ( F 3 ) \left(F3) is presented to restrict the growth of f ( t , x , y , p ) f\left(t,x,y,p) and g ( t , x , y , q ) g\left(t,x,y,q) on p p and q q . The super-linear growth or sub-linear growth of the nonlinear terms f f and g g is described by related inequality conditions instead of the usual independent inequality conditions about f f and g g . The discussion is based on the fixed point index theory in cones.

Existence and approximate controllability results for second-order impulsive stochastic neutral differential systems

This paper focuses on the existence and approximate controllability of second-order non-autonomous impulsive stochastic neutral differential systems in Hilbert spaces. First, by using Schauder's fixed-point theorem, stochastic analysis theory, and evolution operators, a new set of sufficient conditions are formulated, and we prove the existence of mild solutions of second-order non-autonomous impulsive stochastic neutral differential systems. Further, the result is extended to study the approximate controllability results for second-order non-autonomous impulsive stochastic neutral differential systems. Then, a set of sufficient conditions are derived for the approximate controllability of second-order non-autonomous impulsive stochastic neutral differential system by assuming the associated linear system is approximately controllable. The results are obtained with the help of Krasnoselskii's fixed point theorem. Finally, an example is given to illustrate our obtained results.

О РЕАЛИЗАЦИИ ПРИНЦИПА СУПЕРПОЗИЦИИДЛЯ КОНЕЧНОГО ПУЧКА ИНТЕГРАЛЬНЫХ КРИВЫХ БИЛИНЕЙНОЙ СИСТЕМЫ ВТОРОГО ПОРЯДКА

For a finite family of controllable integral curves of the "trajectory, software-positional control" type induced in a separable Hilbert space by a certain second-order bilinear unsteady differential system (for example, hyperbolic), but with different bilinear regulators (with different operator coefficients with a single bilinear form of these regulators), we investigated solvability of the implementation of operator functions of an invariant linear regulator, in the presence of which (in the structure of this differential system) the union of these integral curves represents a family of its admissible solutions. The study is based on the analysis of continuity and semi-additivity of the nonlinear Rayleigh-Ritz functional operator. A numerical illustrative example is given.

State estimation of reaction-diffusion inertial neural networks with Markovian switching channels via conjunct measurement

This paper addresses the state estimation of reaction-diffusion inertial neural networks (RDINNs) with Markovian switching channels based on conjunct measurement method. First, a suitable variable substitution is employed to transform the original second-order differential system into a first-order one, and a time-space sampled-data measurement strategy is adopted to replace the traditional continuous measurement method. As single-channel transmission may suffer from various communication deficiencies, a multi-channel switching scheme is considered in this paper to improve transmission efficiency. Then, by constructing Lyapunov-Krasovskii functional with relaxed conditions, sufficient conditions are derived to ensure that the error system is asymptotically stable. Finally, the feasibility of the proposed method is further validated by simulation results.

The Existence of Periodic Solutions for Second-Order Delay Differential Systems.

In this paper, we consider a kind of second-order delay differential system. By taking some transforms, the property of delay is reflected in the boundary condition. The wonder is that the corrseponding first-order system is exactly the so-called P-boundary value problem of Hamiltonian system which has been studied deeply by many mathematicians, including the authors of this paper. Firstly, we define the relative Morse index for the delay system and give the relationship with the P-index of Hamiltonian system. Secondly, by this index, topology degree and saddle point reduction, the existence of periodic solutions is established for this kind of delay differential system.

A Crank–Nicolson Compact Difference Method for Time-Fractional Damped Plate Vibration Equations

This paper discusses the Crank–Nicolson compact difference method for the time-fractional damped plate vibration problems. For the time-fractional damped plate vibration equations, we introduce the second-order space derivative and the first-order time derivative to convert fourth-order differential equations into second-order differential equation systems. We discretize the space derivative via compact difference and approximate the time-integer-order derivative and fraction-order derivative via central difference and L1 interpolation, respectively, to obtain the compact difference formats with fourth-order space precision and 3−α(1<α<2)-order time precision. We apply the energy method to analyze the stability and convergence of this difference format. We provide numerical cases, which not only validate the convergence order and feasibility of the given difference format, but also simulate the influence of the damping coefficient on the amplitude of plate vibration.

Second-order impulsive differential systems of mixed type: oscillation theorems

In this paper, we obtain necessary and sufficient conditions for the oscillation of solutions to a second-order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is Lebesgue’s dominated convergence theorem.