We establish integral formulas of Minkowski’s type for compact spacelike hypersurfaces in de sitter space S<sub>1</sub><sup>n+1</sup>（1）and give their applications to the case of constant r-...We establish integral formulas of Minkowski’s type for compact spacelike hypersurfaces in de sitter space S<sub>1</sub><sup>n+1</sup>（1）and give their applications to the case of constant r-th mean curvature （r=1,2,…,n-1）.When r=1 we recover Montiel’s result.展开更多
This paper addresses the Singular Optimal Control Problem(SOCP)for a surface-to-air missile with limited control,fully considering aerodynamic effects with a parabolic drag polar.This problem is an extension of the ty...This paper addresses the Singular Optimal Control Problem(SOCP)for a surface-to-air missile with limited control,fully considering aerodynamic effects with a parabolic drag polar.This problem is an extension of the typical Goddard problem.First,the classical Legendre-Clebsch condition is applied to derive optimal conditions for the singular angle of attack,revealing that the missile turns by gravity along the singular arc.Second,the higher-order differentiation of the switching function provides the necessary conditions to determine the optimal thrust,expressed as linear functions of the costate variables.The vanishing coefficient determinant is then employed to decouple the control and costate variables,yielding the singular thrust solely dependent on state variables and identifying the singular surface.Moreover,the analytical singular control can be regarded as path constraints subject to the typical Optimal Control Problem(OCP),enabling the GPOPS-Ⅱ,a direct method framework that does not involve the singular condition,to solve the SOCP.Finally,three cases with different structures are presented to evaluate the performance of the proposed method.The results show that it takes a few steps to obtain the numerical optimal solution,which is consistent with the analytical solution derived from the calculus of variations,highlighting its great computational accuracy and effectiveness.展开更多
A heavy rainfall in the Meiyu front during 4-5 July 2003 is simulated by use of the non-hydrostatic mesoscale model MM5 （V3-6） with different explicit cloud microphysical parameterization schemes. The characteristic...A heavy rainfall in the Meiyu front during 4-5 July 2003 is simulated by use of the non-hydrostatic mesoscale model MM5 （V3-6） with different explicit cloud microphysical parameterization schemes. The characteristics of microphysical process of convective cloud are studied by the model outputs. The simulation study reveals that： （1） The mesoscale model MM5 with explicit cloud microphysical process is capable of simulating the instant heavy rainfall in the Meiyu front, the rainfall simulation could be improved signifi- cantly as the model resolution is increased, and the Goddard scheme is better than the Reisner or Schultz scheme. （2） The convective cloud in the Meiyu front has a comprehensive structure composed of solid, liquid and vapor phases of water, the mass density of water vapor is the largest one in the cloud; the next one is graupel, while those of ice, snow, rain water and the cloud water are almost same. The height at which mass density peaks for different hydrometeors is almost unchangeable during the heavy rainfall period. The mass density variation of rain water, ice, and graupel are consistent with that of ground precipitation, while that of water vapor in the low levels is 1-2 h earlier than the precipitation. （3） The main contribution to the water vapor budget in the atmosphere is the convergence of vapor flux through advection and convection, which provides the main vapor source of the rainfall. Besides the basic process of the auto-conversion of cloud water to rain water, there is an additional cloud microphysical process that is essential to the formation of instant heavy rainfall, the ice-phase crystals are transformed into graupels first and then the increased graupels mix with cloud water and accelerates the conversion of cloud water to rain water. The positive feedback mechanism between latent heat release and convection is the main cause to maintain and develop the heavy precipitation.展开更多
基金Li Haizhong is supported by NNSFC No.19701017 Basic Science Research Foundation of Tsinghua University Chen Weihua is supported by NNSFC No.19571005
文摘We establish integral formulas of Minkowski’s type for compact spacelike hypersurfaces in de sitter space S<sub>1</sub><sup>n+1</sup>（1）and give their applications to the case of constant r-th mean curvature （r=1,2,…,n-1）.When r=1 we recover Montiel’s result.
基金co-supported by the National Natural Science Foundation of China(No.62003019)the Young Talents Support Program of Beihang University,China(No.YWF21-BJ-J-1180)。
文摘This paper addresses the Singular Optimal Control Problem(SOCP)for a surface-to-air missile with limited control,fully considering aerodynamic effects with a parabolic drag polar.This problem is an extension of the typical Goddard problem.First,the classical Legendre-Clebsch condition is applied to derive optimal conditions for the singular angle of attack,revealing that the missile turns by gravity along the singular arc.Second,the higher-order differentiation of the switching function provides the necessary conditions to determine the optimal thrust,expressed as linear functions of the costate variables.The vanishing coefficient determinant is then employed to decouple the control and costate variables,yielding the singular thrust solely dependent on state variables and identifying the singular surface.Moreover,the analytical singular control can be regarded as path constraints subject to the typical Optimal Control Problem(OCP),enabling the GPOPS-Ⅱ,a direct method framework that does not involve the singular condition,to solve the SOCP.Finally,three cases with different structures are presented to evaluate the performance of the proposed method.The results show that it takes a few steps to obtain the numerical optimal solution,which is consistent with the analytical solution derived from the calculus of variations,highlighting its great computational accuracy and effectiveness.
基金Supported by the National Basic Research Program of China under Grant No.2006CB400505the National Natural Science Foundation of China under Grant No.40675040.
文摘A heavy rainfall in the Meiyu front during 4-5 July 2003 is simulated by use of the non-hydrostatic mesoscale model MM5 （V3-6） with different explicit cloud microphysical parameterization schemes. The characteristics of microphysical process of convective cloud are studied by the model outputs. The simulation study reveals that： （1） The mesoscale model MM5 with explicit cloud microphysical process is capable of simulating the instant heavy rainfall in the Meiyu front, the rainfall simulation could be improved signifi- cantly as the model resolution is increased, and the Goddard scheme is better than the Reisner or Schultz scheme. （2） The convective cloud in the Meiyu front has a comprehensive structure composed of solid, liquid and vapor phases of water, the mass density of water vapor is the largest one in the cloud; the next one is graupel, while those of ice, snow, rain water and the cloud water are almost same. The height at which mass density peaks for different hydrometeors is almost unchangeable during the heavy rainfall period. The mass density variation of rain water, ice, and graupel are consistent with that of ground precipitation, while that of water vapor in the low levels is 1-2 h earlier than the precipitation. （3） The main contribution to the water vapor budget in the atmosphere is the convergence of vapor flux through advection and convection, which provides the main vapor source of the rainfall. Besides the basic process of the auto-conversion of cloud water to rain water, there is an additional cloud microphysical process that is essential to the formation of instant heavy rainfall, the ice-phase crystals are transformed into graupels first and then the increased graupels mix with cloud water and accelerates the conversion of cloud water to rain water. The positive feedback mechanism between latent heat release and convection is the main cause to maintain and develop the heavy precipitation.
