In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li C...In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li Chun-hong.展开更多
In this paper. we investigate the growth of solutions of the second-order linear homogeneons differential equations with entire coefficients of the same order. and obtain precise estimates of the hyper-order of their ...In this paper. we investigate the growth of solutions of the second-order linear homogeneons differential equations with entire coefficients of the same order. and obtain precise estimates of the hyper-order of their solutions.展开更多
This paper investigates the growth of solutions of the equation f' + e -zf' + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)ebz, h(z) is nonzero polynomial, b ≠ -1 is a complex constant, every solution o...This paper investigates the growth of solutions of the equation f' + e -zf' + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)ebz, h(z) is nonzero polynomial, b ≠ -1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.展开更多
In this paper, we investigate the growth and fixed points of solutions and their 1st, 2nd derivatives, differential polynomial of second order linear differential equations with meromorphic coefficients, and obtain th...In this paper, we investigate the growth and fixed points of solutions and their 1st, 2nd derivatives, differential polynomial of second order linear differential equations with meromorphic coefficients, and obtain that the exponents of convergence of these fixed points are all equal to the order of growth.展开更多
In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the questi...In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.展开更多
For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit mea...For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.展开更多
The asymptotic behavior and oscillation of the solutions of second order integro-differential equations with deviating argumentis studied. Our technique depends on an integral inequality containing a deviating argumen...The asymptotic behavior and oscillation of the solutions of second order integro-differential equations with deviating argumentis studied. Our technique depends on an integral inequality containing a deviating argument. From this we obtain some sufficient conditions under which all solutions of Eq.(1.4) have some asymptotic behavior and oscillation.展开更多
In this paper, the existence and uniqueness theorems of solutions of k-point boundary value problems for nth-order nonlinear differential equations are established by Leray-Schauder continuation theorem.
This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discret...This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.展开更多
A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous f...A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].展开更多
In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermit...In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.展开更多
In this paper,we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution.We assume that the inhomogeneity appearing in the right-hand term of thi...In this paper,we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution.We assume that the inhomogeneity appearing in the right-hand term of this equation belongs to some anistropic Holder spaces.We illustrate our results by a BVP involving a 3D Laplacian posed in a cusp domain of R^(4).展开更多
In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solv...In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method.In particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing literature.Finally,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.展开更多
A typical adaptive neural control methodology is used for the rigid body model of the hypersonic vehicle. The rigid body model is divided into the altitude subsystem and the velocity subsystem. The proportional integr...A typical adaptive neural control methodology is used for the rigid body model of the hypersonic vehicle. The rigid body model is divided into the altitude subsystem and the velocity subsystem. The proportional integral differential(PID) controller is introduced to control the velocity track. The backstepping design is applied for constructing the controllers for the altitude subsystem.To avoid the explosion of differentiation from backstepping, the higher-order filter dynamic is used for replacing the virtual controller in the backstepping design steps. In the design procedure,the radial basis function(RBF) neural network is investigated to approximate the unknown nonlinear functions in the system dynamic of the hypersonic vehicle. The simulations show the effectiveness of the design method.展开更多
基金This work is supported by the National Natural Science Foundation of China(10161006)the Natural Science Foundation of Jiangxi Prov(001109)Korea Research Foundation Grant(KRF-2001-015-DP0015)
文摘In this paper, authors investigate the order of growth and the hyper order of solutions of a class of the higher order linear differential equation, and improve results of M. Ozawa, G. Gundersen and J.K. Langley, Li Chun-hong.
基金This work is supported by the National Natural Science Foundation of China
文摘In this paper. we investigate the growth of solutions of the second-order linear homogeneons differential equations with entire coefficients of the same order. and obtain precise estimates of the hyper-order of their solutions.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10161006) the Natural Science Foundation of Jiangxi Province.
文摘This paper investigates the growth of solutions of the equation f' + e -zf' + Q(z)f = 0 where the order (Q) = 1. When Q(z) = h(z)ebz, h(z) is nonzero polynomial, b ≠ -1 is a complex constant, every solution of the above equation has infinite order and the hyper-order 1. We improve the results of M. Frei, M. Ozawa, G. Gundersen and J. K. Langley.
基金the National Natural Science Foundation of China(No.10161006)the Natural Science Foundation of Guangdong Province in China(No.04010360)the Brain Pool Program of the Korean Federation of Science and Technology Societies(No.021-1-9)
文摘In this paper, we investigate the growth and fixed points of solutions and their 1st, 2nd derivatives, differential polynomial of second order linear differential equations with meromorphic coefficients, and obtain that the exponents of convergence of these fixed points are all equal to the order of growth.
基金This work was supported by the Brain Pool Program of Korea Federation of Science and Technology Societies (No. 072-1-3-0164)NURI Academy of Banking, Derivatives and Seeurites and Insurance, and the Natural Science Foundation of Guangdong Province in China (Grant No. 06025059)
文摘In this paper, we investigate the existence and the form of subnormal solution for a class of second order periodic linear differential equations, estimate the growth properties of all solutions, and answer the question raised by Gundersen and Steinbart.
基金The author thanks S.Kotani for introducing[7]and[9]to him and R.O˘ınarov for sending him the original version of[12].Thanks are also given to H.J.Zhang and Z.W.Liao for their corrections of an earlier version of the paper.Research supported in part by the National Natural Science Foundation of China(No.11131003)the“985”project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators.Interestingly,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context.All of them are illustrated by examples.The main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.
文摘The asymptotic behavior and oscillation of the solutions of second order integro-differential equations with deviating argumentis studied. Our technique depends on an integral inequality containing a deviating argument. From this we obtain some sufficient conditions under which all solutions of Eq.(1.4) have some asymptotic behavior and oscillation.
文摘In this paper, the existence and uniqueness theorems of solutions of k-point boundary value problems for nth-order nonlinear differential equations are established by Leray-Schauder continuation theorem.
基金supported by the National Natural Science Foundation of China under Grant Nos.61821004 and 62250056the Natural Science Foundation of Shandong Province under Grant Nos.ZR2021ZD14 and ZR2021JQ24+1 种基金Science and Technology Project of Qingdao West Coast New Area under Grant Nos.2019-32,2020-20,2020-1-4,High-level Talent Team Project of Qingdao West Coast New Area under Grant No.RCTDJC-2019-05Key Research and Development Program of Shandong Province under Grant No.2020CXGC01208.
文摘This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.
文摘A new criterion is established for the oscillation of second order superlinear ordinary differential equations of the formx″(t) + p(t)x′(t) + q(t)｜x(t)｜αsgnx(t) = 0, t ≥ t0,where α>1,p and q are continuous functions on[t0,∞). This criterion extends and unifies some of the results obtained in [1]- [5].
文摘In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.
文摘In this paper,we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution.We assume that the inhomogeneity appearing in the right-hand term of this equation belongs to some anistropic Holder spaces.We illustrate our results by a BVP involving a 3D Laplacian posed in a cusp domain of R^(4).
基金supported by the National Natural Science Foundation of China(No.12071403)the Scientific Research Foundation of Hunan Provincial Education Department of China(No.18A049).
文摘In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method.In particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing literature.Finally,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.
基金supported by the National Natural Science Foundation of China (61903374)。
文摘A typical adaptive neural control methodology is used for the rigid body model of the hypersonic vehicle. The rigid body model is divided into the altitude subsystem and the velocity subsystem. The proportional integral differential(PID) controller is introduced to control the velocity track. The backstepping design is applied for constructing the controllers for the altitude subsystem.To avoid the explosion of differentiation from backstepping, the higher-order filter dynamic is used for replacing the virtual controller in the backstepping design steps. In the design procedure,the radial basis function(RBF) neural network is investigated to approximate the unknown nonlinear functions in the system dynamic of the hypersonic vehicle. The simulations show the effectiveness of the design method.
