In this paper,we study the di erence equation a1(z)f(z+1)+a0(z)f(z)=0;where a1(z)and a0(z)are entire functions of nite order.Under some conditions,we obtain some properties,such as xed points,zeros etc.,of the di eren...In this paper,we study the di erence equation a1(z)f(z+1)+a0(z)f(z)=0;where a1(z)and a0(z)are entire functions of nite order.Under some conditions,we obtain some properties,such as xed points,zeros etc.,of the di erences and forward di erences of meromorphic solutions of the above equation.展开更多
This paper presents an experimental investigation of the circulation of the horseshoe vortex system within the equilibrium scour hole at a circular pier, with the data measured by an acoustic Doppler velocimeter （ADV...This paper presents an experimental investigation of the circulation of the horseshoe vortex system within the equilibrium scour hole at a circular pier, with the data measured by an acoustic Doppler velocimeter （ADV）. Velocity vector plots and vorticity contours of the flow field on the upstream plane of symmetry （y = 0 cm） and on the planes ：e3 cm away from the plane of symmetry Cv = ~3 cm） are presented. The vorticity and circulation of the horseshoe vortices were determined using the forward difference technique and Stokes theorem, respectively. The results show that the magnitudes of circulations are similar on the planes y = 3 cm and y = -3 cm, which are less than those on the plane y = 0 cm. The circulation decreases with the increase of flow shallowness, and increases with the densimetric Froude number. It also increases with the pier Reynolds number at a constant densimetric Froude number, or at a constant flow shallowness. The relative vortex strength （dimensionless circulation） decreases with the increase of the pier Reynolds number. Some empirical equations are proposed based on the results. The predicted circulation values with these equations match the measured data, which indicates that these equations can be used to estimate the circulation in future studies.展开更多
This paper considers the linear-quadratic(LQ)optimal control problem for systems governed by a class of second-order parabolic partial differential equations(PDEs).This problem is significant in many areas of mathemat...This paper considers the linear-quadratic(LQ)optimal control problem for systems governed by a class of second-order parabolic partial differential equations(PDEs).This problem is significant in many areas of mathematics,control,engineering and so on,and thus has received great attention in the past decades.Different from the previous articles where the operator is applied to present the controller,the main contribution of this paper is to propose the discretisation-then-continuousization method,which is explicit and implementable.The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations,and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.展开更多
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
In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the lit...In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.展开更多
基金National Natural Science Foundation of China(11801110,11771090,11761035,11871260).
文摘In this paper,we study the di erence equation a1(z)f(z+1)+a0(z)f(z)=0;where a1(z)and a0(z)are entire functions of nite order.Under some conditions,we obtain some properties,such as xed points,zeros etc.,of the di erences and forward di erences of meromorphic solutions of the above equation.
文摘This paper presents an experimental investigation of the circulation of the horseshoe vortex system within the equilibrium scour hole at a circular pier, with the data measured by an acoustic Doppler velocimeter （ADV）. Velocity vector plots and vorticity contours of the flow field on the upstream plane of symmetry （y = 0 cm） and on the planes ：e3 cm away from the plane of symmetry Cv = ~3 cm） are presented. The vorticity and circulation of the horseshoe vortices were determined using the forward difference technique and Stokes theorem, respectively. The results show that the magnitudes of circulations are similar on the planes y = 3 cm and y = -3 cm, which are less than those on the plane y = 0 cm. The circulation decreases with the increase of flow shallowness, and increases with the densimetric Froude number. It also increases with the pier Reynolds number at a constant densimetric Froude number, or at a constant flow shallowness. The relative vortex strength （dimensionless circulation） decreases with the increase of the pier Reynolds number. Some empirical equations are proposed based on the results. The predicted circulation values with these equations match the measured data, which indicates that these equations can be used to estimate the circulation in future studies.
基金supported by the National Natural Science Foundation of China[grant numbers 61821004,62250056]the Natural Science Foundation of Shandong Province[grant numbers ZR2021ZD14,ZR2021JQ24]+2 种基金Science and Technology Project of Qingdao West Coast New Area[grant numbers 2019-32,2020-20,2020-1-4]High-level Talent Team Project of Qingdao West Coast New Area[grant number RCTD-JC-2019-05]Key Research and Development Program of Shandong Province[grant number 2020CXGC01208].
文摘This paper considers the linear-quadratic(LQ)optimal control problem for systems governed by a class of second-order parabolic partial differential equations(PDEs).This problem is significant in many areas of mathematics,control,engineering and so on,and thus has received great attention in the past decades.Different from the previous articles where the operator is applied to present the controller,the main contribution of this paper is to propose the discretisation-then-continuousization method,which is explicit and implementable.The solvability condition of the LQ optimal control problems is given based on a set of differential Riccati equations,and an explicit numerical calculation way of these equations and the design of the optimal controller are provided.
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.
基金supported by NNSF of China(11571090)GCCHB(GCC2014052)
文摘In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.
