SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

1,3-取代方酸衍生物非线性光学性质的从头算和半经验方法的比较研究

The structures of 1,3-substituted squaraine derivatives Sq1 ～ Sq12 were fully optimized by ab initio HF method with 6-31G* basis set level,and the electronic structures of Sq1 ～ Sq12 were also calculated. Based on the optimized structures,the electronic spectra were obtained by the CIS / 6-31G* method,which suggested that the max absorption wavelength mainly resulted from the electronic transition from HOMO to LUMO. The second nonlinear optical coefficients（β0）were calculated using ab initio CPHF method at 6-31G* basis set level and FF / AM1,FF / PM3,FF / MNDO,FF / MINDO3 methods. A systematic comparison between the results was carried out. It indicated that the second nonlinear optical coefficients were affected dramatically by the properties of five-membered hetero-rings. β0 can be enhanced by introducing pyrrole,thiazole and oxazole. The position of five-membered rings containing two heteroatoms which were connected with four-membered squaraine rings also affected β0 .     

二氟亚烷基卡宾与甲醛环加成反应机理的理论研究

用二阶微扰理论研究单重态二氟亚烷基卡宾与甲醛发生的环加成反应机理,采用MP2/6-31G^*方法计算了势能面上各驻点的构型参数、振动频率和能量.结果表明,单重态二氟亚烷基卡宾与甲醛的环加成反应主要有两种反应通道,通道1中,两个反应物经a,b和c三条反应途径生成三元环构型的产物P1,其中途径c是主反应途径,该途径有两步组成:(Ⅰ)二氟亚烷基卡宾与甲醛生成了1个富能中间体(INT1c),是无势垒放热反应,放出能量为219.18kJ/mol;(Ⅱ)中间体(INT1c)异构化为产物二氟亚烷基环氧乙烷,其势垒为134.71kJ/mol.通道2的反应途径由三步组成:(Ⅰ)反应物首先生成了1个富能中间体(INT1b),为无势垒的放热反应,放出的能量142.77kJ/mol;(Ⅱ)中间体(INT1b)异构化成另一中间体(INT2),其势垒为22.31kJ/mol;(Ⅲ)中间体(INT2)异构化成四元环构型产物P2,其势垒为11.98kJ/mol.     

一类奇异次线性两点边值问题的正解

考察了二阶边值问题的正解存在性,其中允许h(t)在t=0,t=1处奇异并允许f(s)在s=0处奇异.     

Study on second harmonic generation of 9-benzylidene-substituted-10-methyl-9,10-dihydroacridines

Eight 9-benzylidene-substituted-10-methyl-9,10-dihydroacridine derivatives were synthesized from acridine as starting material and were characterized by 1H-NMR, 13C-NMR, Ms and elemental analysis. The second harmonic generation (SHG) values of these compounds were determined in powder using Nd:YAG as a laser source, as compared with urea powder, and the values of second-order polarizabilities (betaxxx), the values of the composite magnitude (betaCTmicrog) of molecular hypersusceptibilities and their moment of these compounds were obtained by the solvatochromic method under ground state for everyone. The results showed that SHG value of 10 is higher than that of urea; the betaCTmicrog of 5 (107.8 x 10(-30) esu) is lower than that of 4-nitro-N,N-dimethylaniline (30 x 10(-30) esu); the betaCTmicrog of 7 (350.8 x 10(-30) esu) and 10 (244.6 x 10(-30) esu) are higher than that of 4-nitro-N,N-dimethylaniline; the betaCTmicrog of 8 (3553 x 10(-30) esu), 11 (1187 x 10(-30) esu) and 12 (1163 x 10(-30) esu) are much more higher than that of 4-nitro-N,N-dimethylaniline. The results demonstrated that this series of compounds possesses good second-order nonlinear optical (NLO) property. The regular relationship could not be obtained between electronegativity of substituents (R) attached to benzylidene ring and SHG values or betaCTmicrog values although the Rs are different in electronegativity and should make an effect on the extent of intramolecular electron-transfer and would consequently influence SHG or betaCTmicrog. The electron-withdrawing ability of R from benzylidene ring played an important role on lambdamax of these compounds.     

十二顶点邻位双取代碳硼烷衍生物二阶NLO性质的理论研究

采用密度泛函理论(DFT) B3LYP/6-31G*方法, 对系列十二顶点邻位双取代碳硼烷(C2B10H12)衍生物的几何构型进行优化. 在所得优化结构的基础上, 结合有限场方法(FF)和含时密度泛函理论(TD-DFT)对这些分子的二阶非线性光学(NLO)活性及电子吸收光谱进行了研究. 结果表明, 邻位双取代碳硼烷有较强的吸电子作用, 与有机基团形成D-π-A结构时, 可以起到很好的受体作用. 当给体部分或桥的共轭性好, 给体的给电子能力强时, 邻位双取代碳硼烷的吸电子作用更明显, 从而增强了分子的二阶NLO响应.     

The covering radius of the Reed–Muller code RM(2,7) is 40

It was proved by J. Schatz that the covering radius of the second order Reed–Muller code $RM(2,6)$ is 18 (Schatz (1981)). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for the covering radius of $RM(2,n)$, $n=8,9,10$.     

Oscillation and Nonoscillation Criteria for Second-Order Linear Differential Equations

Sufficient conditions for oscillation and nonoscillation of second-order linear equations are established.     

二次锥规划的光滑牛顿法

在光滑Fischer-Burmeister函数的基础上,本文给出了二次锥规划的一种新的光滑牛顿法.该方法所采用的系统不是等价于中心路径条件,而是等价于最优性条件本身.算法对初始点没有任何限制,且具有Q-二阶收敛速度.