This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–M...This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero.展开更多
The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the po...The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.展开更多
An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditio...An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.展开更多
The asymptotic and stable properties of general stochastic functional differential equations are investigated by the multiple Lyapunov function method, which admits non-negative up-per bounds for the stochastic deriva...The asymptotic and stable properties of general stochastic functional differential equations are investigated by the multiple Lyapunov function method, which admits non-negative up-per bounds for the stochastic derivatives of the Lyapunov functions, a theorem for asymptotic properties of the LaSal e-type described by limit sets of the solutions of the equations is obtained. Based on the asymptotic properties to the limit set, a theorem of asymptotic stability of the stochastic functional differential equations is also established, which enables us to construct the Lyapunov functions more easily in application. Particularly, the wel-known classical theorem on stochastic stability is a special case of our result, the operator LV is not required to be negative which is more general to fulfil and the stochastic perturbation plays an important role in it. These show clearly the improvement of the traditional method to find the Lyapunov functions. A numerical simulation example is given to il ustrate the usage of the method.展开更多
This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solutio...This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.展开更多
In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarant...In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarantee of computations with a given precision. The equations of programmed constraints and those of constraint perturbations are defined. The stability of the programmed manifold for numerical solutions of the kinematical and dynamical equations is obtained by corresponding construction of the constraint perturbation equations. The dynamical equations of system with programmed constraints are set up in the form of Lagrange’s equations in generalized coordinates. Certain inverse problems of rigid body dynamics are examined.展开更多
A backstepping method is used for nonlinear spacecraft attitude stabilization in the presence of external disturbances and time delay induced by the actuator. The kinematic model is established based on modified Rodri...A backstepping method is used for nonlinear spacecraft attitude stabilization in the presence of external disturbances and time delay induced by the actuator. The kinematic model is established based on modified Rodrigues parameters (MRPs). Firstly, we get the desired angular velocity virtually drives the attitude parameters to origin, and then backstep it to the desired control torque required for stabilization. Considering the time delay induced by the actuator, the control torque functions only after the delayed time, therefore time compensation is needed in the controller. Stability analysis of the close-loop system is given afterwards. The infinite dimensional actuator state is modeled with a first-order hyperbolic partial differential equation (PDE), the L-2 norm of the system state is constructed and is proved to be exponentially stable. An inverse optimality theorem is also employed during controller design. Simulation results illustrate the efficiency of the proposed control law and it is robust to bounded external disturbances and time delay mismatch.展开更多
In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic ...In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.展开更多
The change of stress and temperature in the conform forming process has been studied. On the basis of the law of momentum, the law of momantum moment and the law of energy conservation, the governmental differential e...The change of stress and temperature in the conform forming process has been studied. On the basis of the law of momentum, the law of momantum moment and the law of energy conservation, the governmental differential equations of stress and temperature in the conform metal forming process have been derived, whose definite conditions are given by material mechanics, elastic machanics and plastic mechanics. The analytic solution of these equations has been successfully obtained and as a result, the solid foundations and scientific guide for the further development of this prospective metal processing techique have been established.展开更多
A new approach is proposed to model nonlinear dynamic systems by combining SOM(self-organizing feature map) with support vector regression (SVR) based on expert system. Thewhole system has a two-stage neural network a...A new approach is proposed to model nonlinear dynamic systems by combining SOM(self-organizing feature map) with support vector regression (SVR) based on expert system. Thewhole system has a two-stage neural network architecture. In the first stage SOM is used as a clus-tering algorithm to partition the whole input space into several disjointed regions. A hierarchicalarchitecture is adopted in the partition to avoid the problem of predetermining the number of parti-tioned regions. Then, in the second stage, multiple SVR, also called SVR experts, that best fit eachpartitioned region by the combination of di?erent kernel function of SVR and promote the configura-tion and tuning of SVR. Finally, to apply this new approach to time-series prediction problems basedon the Mackey-Glass di?erential equation and Santa Fe data, the results show that SVR experts hase?ective improvement in the generalization performance in comparison with the single SVR model.展开更多
Error analysis methods in frequency domain are developed in this paper for determining the characteristic root and transfer function errors when the linear multipass algorithms are used to solve linear differential eq...Error analysis methods in frequency domain are developed in this paper for determining the characteristic root and transfer function errors when the linear multipass algorithms are used to solve linear differential equations. The relation between the local truncation error in time domain and the error in frequency domain is established, which is the basis for developing the error estimation methods. The error estimation methods for the digital simulation model constructed by using the Runge-Kutta algorithms and the linear multistep predictor-corrector algorithms are also given.展开更多
In this paper, the stability analysis for parallel real-time digital simulation models is discussed. The coupling coefficient perturbation method and the simulation stepsize perturbation method are established. For tw...In this paper, the stability analysis for parallel real-time digital simulation models is discussed. The coupling coefficient perturbation method and the simulation stepsize perturbation method are established. For two classes of systems of test equations, we construct the parallel simulation models and prove that they have the stability behaviour which is similar to the original continuous systems.展开更多
A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is pr...A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is proved and the stability analysis is given. The numerical examples demonstrate that these formulas can solve an extensive class of initial value problems for the ordinary differential equations.展开更多
基金supported by the PhD Research Startup Foundation of Hubei University of Economics(Grand No.XJ23BS42).
文摘This paper studies the Smoluchowski–Kramers approximation for a discrete-time dynamical system modeled as the motion of a particle in a force field.We show that the approximation holds for the drift-implicit Euler–Maruyama discretization and derive its convergence rate.In particular,the solution of the discretized system converges to the solution of the first-order limit equation in the mean-square sense,and this convergence is independent of the order in which the mass parameterμand the step size h tend to zero.
基金supported by the National Board for Higher Mathematics,Mumbai,India under Grant No.2/48(5)/2013/NBHM(R.P.)/RD-II/688 dt 16.01.2014
文摘The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.
文摘An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods （TSCRK） is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient.
基金supported by the National Natural Science Foundation of China(61273126)the Natural Science Foundation of Guangdong Province(10251064101000008+1 种基金S201210009675)the Fundamental Research Funds for the Central Universities(2012ZM0059)
文摘The asymptotic and stable properties of general stochastic functional differential equations are investigated by the multiple Lyapunov function method, which admits non-negative up-per bounds for the stochastic derivatives of the Lyapunov functions, a theorem for asymptotic properties of the LaSal e-type described by limit sets of the solutions of the equations is obtained. Based on the asymptotic properties to the limit set, a theorem of asymptotic stability of the stochastic functional differential equations is also established, which enables us to construct the Lyapunov functions more easily in application. Particularly, the wel-known classical theorem on stochastic stability is a special case of our result, the operator LV is not required to be negative which is more general to fulfil and the stochastic perturbation plays an important role in it. These show clearly the improvement of the traditional method to find the Lyapunov functions. A numerical simulation example is given to il ustrate the usage of the method.
基金supported by the National Natural Science Foundation of China(61370136)the Hainan Province Science and Technology Cooperation Fund Project(KJHZ2015-36)the Hainan Province Introduced and Integrated Demonstration Projects(YJJC20130009)
文摘This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.
基金Supported by Russian Fund of Fund amental Investigations(Pr.990101064)and Russian Minister of Educatin
文摘In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarantee of computations with a given precision. The equations of programmed constraints and those of constraint perturbations are defined. The stability of the programmed manifold for numerical solutions of the kinematical and dynamical equations is obtained by corresponding construction of the constraint perturbation equations. The dynamical equations of system with programmed constraints are set up in the form of Lagrange’s equations in generalized coordinates. Certain inverse problems of rigid body dynamics are examined.
文摘A backstepping method is used for nonlinear spacecraft attitude stabilization in the presence of external disturbances and time delay induced by the actuator. The kinematic model is established based on modified Rodrigues parameters (MRPs). Firstly, we get the desired angular velocity virtually drives the attitude parameters to origin, and then backstep it to the desired control torque required for stabilization. Considering the time delay induced by the actuator, the control torque functions only after the delayed time, therefore time compensation is needed in the controller. Stability analysis of the close-loop system is given afterwards. The infinite dimensional actuator state is modeled with a first-order hyperbolic partial differential equation (PDE), the L-2 norm of the system state is constructed and is proved to be exponentially stable. An inverse optimality theorem is also employed during controller design. Simulation results illustrate the efficiency of the proposed control law and it is robust to bounded external disturbances and time delay mismatch.
基金The project was supported by the National Natural Science Faundation of China
文摘In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.
文摘The change of stress and temperature in the conform forming process has been studied. On the basis of the law of momentum, the law of momantum moment and the law of energy conservation, the governmental differential equations of stress and temperature in the conform metal forming process have been derived, whose definite conditions are given by material mechanics, elastic machanics and plastic mechanics. The analytic solution of these equations has been successfully obtained and as a result, the solid foundations and scientific guide for the further development of this prospective metal processing techique have been established.
基金Supported by the National High Technology Research and Development Program of P.R.China(2002AA412010)the Technology Development Program of the Ministry of Science and Technology of P.R.China(2003EG113016)
文摘A new approach is proposed to model nonlinear dynamic systems by combining SOM(self-organizing feature map) with support vector regression (SVR) based on expert system. Thewhole system has a two-stage neural network architecture. In the first stage SOM is used as a clus-tering algorithm to partition the whole input space into several disjointed regions. A hierarchicalarchitecture is adopted in the partition to avoid the problem of predetermining the number of parti-tioned regions. Then, in the second stage, multiple SVR, also called SVR experts, that best fit eachpartitioned region by the combination of di?erent kernel function of SVR and promote the configura-tion and tuning of SVR. Finally, to apply this new approach to time-series prediction problems basedon the Mackey-Glass di?erential equation and Santa Fe data, the results show that SVR experts hase?ective improvement in the generalization performance in comparison with the single SVR model.
基金This project was supported by the National Natural Science Foundation of China (No. 19871080).
文摘Error analysis methods in frequency domain are developed in this paper for determining the characteristic root and transfer function errors when the linear multipass algorithms are used to solve linear differential equations. The relation between the local truncation error in time domain and the error in frequency domain is established, which is the basis for developing the error estimation methods. The error estimation methods for the digital simulation model constructed by using the Runge-Kutta algorithms and the linear multistep predictor-corrector algorithms are also given.
基金This work is supported partly by the National Natural Science Foundation of China
文摘In this paper, the stability analysis for parallel real-time digital simulation models is discussed. The coupling coefficient perturbation method and the simulation stepsize perturbation method are established. For two classes of systems of test equations, we construct the parallel simulation models and prove that they have the stability behaviour which is similar to the original continuous systems.
基金Project supported by the National Natural Science Foundation of China
文摘A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is proved and the stability analysis is given. The numerical examples demonstrate that these formulas can solve an extensive class of initial value problems for the ordinary differential equations.
