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
The impermeability of isentropic surfaces by the potential vorticity substance （PVS） has often been used to help understand the generation of potential vorticity in the presence of diabatic heating and friction. In ...The impermeability of isentropic surfaces by the potential vorticity substance （PVS） has often been used to help understand the generation of potential vorticity in the presence of diabatic heating and friction. In this study, we examined singularities of isentropic surfaces that may develop in the presence of diabatic heating and the fictitious movements of the isentropic surfaces that are involved in deriving the PVS impermeability theorem. Our results show that such singularities could occur in the upper troposphere as a result of intense convective-scale motion, at the cloud top due to radiative cooling, or within the well-mixed boundary layer. These locally ill-defined conditions allow PVS to penetrate across an isentropic surface. We conclude that the PVS impermeability theorem is generally valid for the stably stratified atmosphere in the absence of diabatic heating.展开更多
This note gives two examples of surfaces with normal crossing singularities.In the first example the canonical ring is not finitely generated.In the second,the canonical line bundle is not ample but its pull back to t...This note gives two examples of surfaces with normal crossing singularities.In the first example the canonical ring is not finitely generated.In the second,the canonical line bundle is not ample but its pull back to the normalization is ample.The latter answers in the negative a problem left unresolved in Ⅲ.2.6.2 of lments de gometrie algbrique,1961,and raised again by Viehweg.展开更多
We consider the boundedness of the rough singular integral operator T_（？,ψ,h） along a surface of revolution on the Triebel-Lizorkin space F^α_（ p,q）（R^n） for Ω ∈ H^1（（S^n-1）） and Ω ∈ Llog^＋L（S^n-1）...We consider the boundedness of the rough singular integral operator T_（？,ψ,h） along a surface of revolution on the Triebel-Lizorkin space F^α_（ p,q）（R^n） for Ω ∈ H^1（（S^n-1）） and Ω ∈ Llog^＋L（S^n-1） ∪_1展开更多
IN this letter we discuss the necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property. According to ref. [1], on a closed surface, ever...IN this letter we discuss the necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property. According to ref. [1], on a closed surface, every minimal set of a C^r(r≥2) flow is trivial, but it is possible for a C^0 flow to contain non-trivial minimal sets. Thus C^0 flows on closed surfaces are more complicated than C^r(r≥2) flows.展开更多
This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= ...This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= 0 in agreement with the Manin-Peyre conjectures.展开更多
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 bythe National Science Foundation (USAGrant No. ATM-0758609)+1 种基金the National Aeronautics and Space Administration (USAGrant No. NNG05GR32G)
文摘The impermeability of isentropic surfaces by the potential vorticity substance （PVS） has often been used to help understand the generation of potential vorticity in the presence of diabatic heating and friction. In this study, we examined singularities of isentropic surfaces that may develop in the presence of diabatic heating and the fictitious movements of the isentropic surfaces that are involved in deriving the PVS impermeability theorem. Our results show that such singularities could occur in the upper troposphere as a result of intense convective-scale motion, at the cloud top due to radiative cooling, or within the well-mixed boundary layer. These locally ill-defined conditions allow PVS to penetrate across an isentropic surface. We conclude that the PVS impermeability theorem is generally valid for the stably stratified atmosphere in the absence of diabatic heating.
基金provided by the NSF under grant number DMS-0500198
文摘This note gives two examples of surfaces with normal crossing singularities.In the first example the canonical ring is not finitely generated.In the second,the canonical line bundle is not ample but its pull back to the normalization is ample.The latter answers in the negative a problem left unresolved in Ⅲ.2.6.2 of lments de gometrie algbrique,1961,and raised again by Viehweg.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371057, 11471033 and 11571160)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20130003110003)+2 种基金the Fundamental Research Funds for the Central Universities (Grant No. 2014KJJCA10)Grant-in-Aid for Scientific Research (C) (Grant No. 23540228)Japan Society for the Promotion of Science
文摘We consider the boundedness of the rough singular integral operator T_（？,ψ,h） along a surface of revolution on the Triebel-Lizorkin space F^α_（ p,q）（R^n） for Ω ∈ H^1（（S^n-1）） and Ω ∈ Llog^＋L（S^n-1） ∪_1
文摘IN this letter we discuss the necessary and sufficient condition of C^0 flows on closed surfaces with isolated singular points having the pseudo-orbit tracing property. According to ref. [1], on a closed surface, every minimal set of a C^r(r≥2) flow is trivial, but it is possible for a C^0 flow to contain non-trivial minimal sets. Thus C^0 flows on closed surfaces are more complicated than C^r(r≥2) flows.
基金supported by the program PRC 1457-Au For Di P(CNRS-NSFC)supported by National Natural Science Foundation of China(Grant No.11531008)+1 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Province
文摘This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= 0 in agreement with the Manin-Peyre conjectures.
