Second-order Differential Systems Research Articles

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

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

Multiple fast homoclinic solutions for a class of second-order differential systems with p -Laplacian

UDC 517.9 We study the existence of multiple fast homoclinic solutions for a class of second-order differential systems with p -Laplacian by using the minimax methods in the critical-point theory.

Existence of positive solutions to impulsive nonlinear differential systems of second order with two point boundary conditions

Existence of positive solutions to impulsive nonlinear differential systems of second order with two point boundary conditions

Existence of solutions and approximate controllability of second-order stochastic differential systems with Poisson jumps and finite delay

Existence of solutions and approximate controllability of second-order stochastic differential systems with Poisson jumps and finite delay

Variational approach to Kirchhoff-type second-order impulsive differential systems

Abstract In this study, we consider a Kirchhoff-type second-order impulsive differential system with the Dirichlet boundary condition and obtain the existence and multiplicity of solutions to the impulsive problem via variational methods.

On the approximate controllability of second-order stochastic differential systems with hemivariational inequalities with damping

In this article, we investigate the approximate controllability of damping second-order stochastic differential systems with hemivariational inequalities. First, we show the existence of a mild solution to the suggested second-order differential system with the notions of generalized Clarke's subdifferential type. Next, by adopting the fixed point technique of condensing multi-valued maps, the conditions for generalized Clarke's subdifferential and the strongly continuous cosine family is employed for the existence of a mild solution. Furthermore, the approximate controllability results for the given system are proven. In the end, an illustration is presented to highlight our primary findings.

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

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

In the present paper, we obtain infinitely many pairs of homoclinic solutions for a class of nonsmooth second-order differential systems when the energy functional associated is not continuously differentiable and does not satisfy the Palais–Smale condition.

Approximate controllability of second-order neutral stochastic differential evolution systems with random impulsive effect and state-dependent delay

<p>In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.</p>

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.