Second-order Differential Systems Research Articles

Results on controllability and well-posedness of functional abstract second-order differential equations with state-dependent delay

The major goal of this study is to demonstrate the existence and controllability of a second-order abstract differential system with state-dependent delay. We also discuss the well-posedness of the problem at hand. Finally, some examples are provided for the illustration of the main results.

Consensus Problem and Formation Control for Heterogeneous Multi-Agent Systems with Switching Topologies

The cooperative control problem of discrete-time multi-agent systems (MASs) is discussed, and bounded uncertain time-delays and directed switching topologies are considered. By applying model transformations and matrix theory, an augmented system method is introduced to handle a heterogeneous time-delay MAS. Then, the consensus problem of the system is turned to the convergence issue of the product of innumerable row stochastic matrices. Sufficient conditions for asymptotic consensus of the system under directed switching topologies are obtained. Moreover, a novel consensus-based formation control strategy is designed to gain sufficient and necessary conditions for the formation control of a second-order differential robot system. Finally, the effectiveness of the obtained results is verified through simulations.

Viscoacoustic wave equation for qP-wave in transversely isotropic media

The wave equation acts a pivotal part in seismic data processing. We derived two time-domain pure qP viscoacoustic wave equations suitable for attenuation anisotropic media, providing a choice for pure qP-wave forward modeling and inversion in attenuation anisotropic media. Our new qP-wave equation in vertical transversely isotropic (VTI) media are based on the dispersion relation for qP-wave in VTI media and obtained by the relaxation functions of the standard linear solid model and the Kjartansson's constant Q model respectively. Anelasticity including amplitude attenuation and phase dispersion can be dealt with by the proposed wave equation separately. With the coordinate transform, the corresponding two pure qP-wave viscoacoustic anisotropic wave equations for titled transversely isotropic media also are derived. Different from the common viscoacoustic anisotropic wave equation (a coupled second-order partial differential equations system), the new wave equations are simplified by decomposing the original pseudo-differential operator into a differential and a scalar opera- tors. For numerical calculation, the new wave equations can be solved more efficiently than the coupled wave equations. Meanwhile the new wave equations could simulate pure qP-wave steadily in attenuation anisotropic media without quasi-SV wave artifacts. Some synthetic examples, including attenuation Hess and Marmousi-2 models, demonstrate the feasibility of the derived pure qP-wave viscoacoustic wave equations in transversely isotropic media.

An analysis concerning approximate controllability results for second-order Sobolev-type delay differential systems with impulses

This paper is devoted to studying the approximate controllability for second-order impulsive differential inclusions with infinite delay. For proving the main results, we use the results related to the cosine and sine function of operators, Martelli’s fixed point theorem, and the results when combined with the properties of differential inclusions. Firstly, we prove the approximate controllability for second-order impulsive differential inclusions with initial conditions. Then, we extend the discussion to the second-order impulsive system with nonlocal conditions. Finally, we provide an example for the illustration of the obtained theoretical results.

Faedo–Galerkin method for impulsive second-order stochastic integro-differential systems

This paper studies impulsive second-order stochastic differential systems in a separable Hilbert space X. By using the projection operators, we restrict the given problem to a finite-dimensional subspace. The existence and convergence of estimated solutions for the considered problem are investigated via the theories of cosine family and fractional powers of a closed linear operator. We also examine the existence and convergence of the Faedo–Galerkin approximate solutions. At last, we are constructed some examples to demonstrate the effectiveness of the obtained results.

The Second-Order Differential Equation System with the Feedback Controls for Solving Convex Programming

The Second-Order Differential Equation System with the Feedback Controls for Solving Convex Programming

ON THE CONTROLLABILITY OF LINEAR EVOLUTIONARY FUNCTIONAL-DIFFERENTIAL SYSTEMS

New theorem on the controllability of linear functional-differential system of evolutionary type and algorithm for practical verification of controllability are obtained, which can be applied even in the case when the coefficients of system are not continuously differentiable over the time interval under consideration. Special cases of this system are nonstationary systems with distributed and concentrated delay, integro-differential systems with a Voltaire integral and ordinary differential systems. The main results obtained are formulated in the form of 12 theorems and 3 corollaries. On their basis an algorithm for practical verification of the controllability of the system under consideration using a computer is built. Examples illustrating the operability of the obtained theorems and the algorithm are given. The algorithm is implemented in the Maple 17 package for examples of second-order differential systems with delay.

On nonlinear impulsive differential systems with canonical and non-canonical operators

The investigation on the oscillatory and asymptotic properties of differential equations plays an important role in many applications, such as engineering, physics and medicine. In this work, we establish some new sufficient conditions for both the oscillatory and asymptotic behavior of the solutions to a second-order neutral impulsive differential system (IDS, for short) with deviating argument and fixed moments of impulse effect under both canonical and non-canonical operators with various values of the neutral coefficient. Also, we obtain the condition for the existence of a positive solution of the system under consideration. Finally, two examples are provided to illustrate the main results.

Finite-time stabilization of memristor-based inertial neural networks with time-varying delays combined with interval matrix method

This paper investigates the finite-time stability problem of memristor-based inertial neural networks (MINNs) with time-varying delays in association with the interval matrix approach. In the light of differential inclusion and set-valued mapping, the delayed MINNs are converted to a category of systems with uncertain terms by means of convex combination transformations of matrices to deal with the problem of the parameter mismatch of MINNs. Afterwards, the second-order differential systems are simplified to the ordinary first-order differential systems utilizing dimension reduction. Then, two types of delayed feedback controllers are designed to ensure that MINNs can achieve finite-time stabilization, which handles time-varying delays in different ways. Meanwhile, the finite-time stabilization criterion can be deduced by linear matrix inequalities (LMIs). Therefore, the upper bound on the settling time is estimated as one of the indicators to measure the stability effect. Finally, the accuracy of the theoretical results and the validity of the proposed method are verified by two examples.

Analysis on Controllability Results for Wellposedness of Impulsive Functional Abstract Second-Order Differential Equation with State-Dependent Delay

The functional abstract second order impulsive differential equation with state dependent delay is studied in this paper. First, we consider a second order system and use a control to determine the controllability result. Then, using Sadovskii’s fixed point theorem, we get sufficient conditions for the controllability of the proposed system in a Banach space. The major goal of this study is to demonstrate the controllability of an abstract second-order impulsive differential system with a state dependent delay mechanism. The wellposed condition is then defined. Next, we studied whether the defined problem is wellposed. Finally, we apply our results to examine the controllability of the second order state dependent delay impulsive equation.

Derivation of a quasi-linear second-order elliptic-parabolic model for the efficiency of silicon solar cells

In this note, we propose a mathematical model for the efficiency of a silicon solar cell. The model is rigorously derived from continuity and transport assumptions, along with the Poisson equations to describe the behavior of charge carriers in a semiconductor. The photovoltaic phenomenon was thoroughly studied to develop the model, and some parameters that play a crucial role in its efficiency have been identified. Those parameters include the diffusion, the mobility and the absorption coefficients, the permittivity, the regions width and the minority carrier lifetime. The mathematical model obtained herein is a quasi-linear second-order elliptic-parabolic partial differential equation system with strong coupling and interface. The model is defined over a spatial domain composed of three regions with different electrical properties. The fundamental physical laws of the photovoltaic effect are employed here to develop the mathematical model and its boundary conditions.

Non-fragile dissipative state estimation for semi-Markov jump inertial neural networks with reaction-diffusion

In this paper, the non-fragile dissipative state estimation is addressed for semi-Markov jump inertial neural networks with reaction-diffusion. A semi-Markov jump model is used to describe the stochastic jump parameters in networks. Different from the invariable transition probabilities in the traditional Markov jump systems, the transition probabilities of the semi-Markov jump systems rely on the stochastic sojourn-time. Accordingly, the Weibull distribution taking the place of the exponential distribution in this paper is adopted for the sojourn-time of each mode in the system. Firstly, by utilizing an applicable vector substitution, the second-order differential system could be converted into the first-order one. Afterwards, via constructing a seemly Lyapunov function of the semi-Markov inertial neural networks and adequately taking advantage of the peculiarities of cumulative distribution functions, some sufficient conditions with less conservatism are constructed to assure that the estimation error system is strictly (R1,R2,R3)−ϱ−dissipative stochastically stable. Based on these conditions, mode-dependent estimator gains are designed. Finally, a numerical example is proposed to validate the availability of the provided approach.

Second-order impulsive differential systems with mixed and several delays

In this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

Design of Spline-Evolutionary Computing Paradigm for Nonlinear Thin Film Flow Model.

A novel design of stochastic numerical computing method is introduced for computational fluid dynamics problem governed with nonlinear thin film flow (TFF) system by exploiting the competency of polynomial splines for discretization and optimization with evolutionary computing aided with brilliance of local search. The TFF model of second grade fluid is represented with nonlinear second-order differential system. The aim of the present work is to exploit the cubic spline approach (CSA) to transform the differential equations for TFF model into an equivalent set of nonlinear equations. The approximation in mean squared error sense is introduced for the formulation of cost function for solving the nonlinear system of equations representing TFF model. The optimization of the decision variables of the cost function is carried out with global search efficacy of evolution by genetic algorithms (GAs) integrated with sequential quadratic programming (SQP) for speedy adjustments. The designed spline–evolutionary computing paradigm, CSA–GA–SQP, is evaluated for different scenarios of TFF model by variation of second grade and magnetic parameters, as well as variation in the length of splines. Results endorsed the worth of CSA–GA–SQP solver as an efficient alternative, reliable, stable, and accurate framework for the variants of nonlinear TFF systems on the basis of multiple autonomous executions. The design computing spline paradigm CSA–GA–SQP is a promising alternative numerical solver to be implemented for the solution of stiff nonlinear systems representing the complex scenarios of computational fluid dynamics problems.

The Second-Order Differential Equation System with the Controlled Process for Variational Inequality with Constraints

In this paper, the variational inequality with constraints can be viewed as an optimization problem. Using Lagrange function and projection operator, the equivalent operator equations for the variational inequality with constraints under the certain conditions are obtained. Then, the second-order differential equation system with the controlled process is established for solving the variational inequality with constraints. We prove that any accumulation point of the trajectory of the second-order differential equation system is a solution to the variational inequality with constraints. In the end, one example with three kinds of different cases by using this differential equation system is solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the controlled process for solving the variational inequality with constraints.

New Results on Qualitative Behavior of Second Order Nonlinear Neutral Impulsive Differential Systems with Canonical and Non-Canonical Conditions

The oscillation of impulsive differential equations plays an important role in many applications in physics, biology and engineering. The symmetry helps to deciding the right way to study oscillatory behavior of solutions of impulsive differential equations. In this work, several sufficient conditions are established for oscillatory or asymptotic behavior of second-order neutral impulsive differential systems for various ranges of the bounded neutral coefficient under the canonical and non-canonical conditions. Here, one can see that if the differential equations is oscillatory (or converges to zero asymptotically), then the discrete equation of similar type do not disturb the oscillatory or asymptotic behavior of the impulsive system, when impulse satisfies the discrete equation. Further, some illustrative examples showing applicability of the new results are included.

Second-Order Impulsive Delay Differential Systems: Necessary and Sufficient Conditions for Oscillatory or Asymptotic Behavior

In this work, we aimed to obtain sufficient and necessary conditions for the oscillatory or asymptotic behavior of an impulsive differential system. It is easy to notice that most works that study the oscillation are concerned only with sufficient conditions and without impulses, so our results extend and complement previous results in the literature. Further, we provide two examples to illustrate the main results.

Rotating periodic integrable solutions for second-order differential systems with nonresonance condition

In this paper, by using Parseval's formula and Schauder's fixed point theorem, we prove the existence and uniqueness of rotating periodic integrable solution of the second-order system $x''+f(t,x)=0$ with $x(t+T)=Qx(t)$ and $\int_{(k-1)T}^{kT}x(s)ds=0$, $k\in Z^+$ for any orthogonal matrix $Q$ when the nonlinearity $f$ satisfies nonresonance condition.

К теории переопределенных систем дифференциальных уравнений в частных производных второго порядка с одной сингулярной линией

In this paper, an over determined system of second-order partial differential equations with one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular line. Under the condition of compatibility, introducing a new function, we come to a over determined system of partial differential equations of the second order with one singular line of a simpler form. The integral representation of the manifold of solutions of the redefined second-order partial differential system with one singular line is found explicitly through three arbitrary constants, for which initial data problems (Cauchy type problems) can be posed.

ON THE EXISTENCE OF A CONDITIONALLY PERIODIC SOLUTION OF A QUASILINEAR SYSTEM DIFFERENTIAL EQUATION IN THE CRITICAL CASE

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.