伴随函数

The Frechet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. The advantage is that the cost function variation is independent of the flow field variation, with the result that the gradient of cost function with respect to arbitrary number of design variables can be determined without the need for additional flow-field evaluations. So each design cycle requires the numerical solution of both the flow and the adjoint equations leading to a computational cost roughly equal to the cost of two flow solutions.

虽然他们的研究成果大多数都可以在公开性文献上看到，但是其中一些具体的细节问题的处理方法却没有透露，比如：伴随方程如何离散求解才能减小最终梯度计算中的误差，使得设计过程有效地进行；伴随方程与流动方程的边界条件是不同的，如何合理地处理伴随方程的边界条件，才能推动设计过程朝着设计目标而发展；在减阻问题中目标函数如何定义，求解梯度的具体数学公式如何推导，如何通过数值方法来实现它们等。

Definite economic explanations to spatial adjoint function and control variables are presented.

并对空间伴随函数和控制变量给出了明确的经济学解释。

And an optimal operation strategy of any kind of products subject to spoilage at any time is obtained on the basis of the proposition of the model of distributed parameter control systems. Moreover, definite economic explanations to spatial adjoint function and control variables are presented.

通过建立可变质产品的最优经营管理的分布参数控制系统模型，得出了零售企业每个瞬间每种可变质产品的最优经营管理策略，并对空间伴随函数和控制变量给出了明确的经济学解释。

Chavent's research for matching the history of production in a field by use of optimal control theory and through the research for the space of the adjoint function, we have extended the theory of adjustment for the ease of a single-phase system to a double-phase one.

Chavent利用最优化理论研究油气藏生产动态拟合的基础上，通过对伴随函数空间的研究，将辨识理论由单重介质系统扩展至双重介质系统，并针对原方法多解，迭代中易发生逸出、振荡、仅收敛于局部极小等困难，重点研究和探讨了改善问题的适定性和加速算法的收敛性问题。

In this paper,a systematic direct perturbation method of dark solitons is found.Having analyzed the mistakes in earlier works on perturbation method for dark solitonsand essence of the direct perturbation method for bright solitons,we notice that to in-troduce the adjoint solutions of the squared Jost solutions and to prove the completenessare crucial to the problem.Giving up the unnecessary scheme of introducing the adjointoperator in the bright soliton case,we directly find the adjoint solutions by meetingthe demand for the orthogonality that inner product of the squared Jost solutions andits adjoint should be proportional to a δ function in the case of continuous spectra.The corresponding adjoint operator is thus found.Taking into account the reductiontransformation,we find a correct description for the completeness of the squared Jostsolutions and directly verify its validity with explicit expressions of the squared Jostsolutions.

本论文建立了系统的暗孤子直接微扰方法，在对前人关于暗孤子微扰方法的错误以及亮孤子直接微扰方法的本质作了充分的分析后，认识到引入平方Jost解的伴随解和证明完备性是问题的关键，撇开过去亮孤子情况首先引入伴随算子的非必要作法，直接从平方Jost解与其伴随解的内积在连续谱时正比于δ函数这一正交性要求出发，找出了伴随解，同时得出了应有的伴随算子，在考虑到约化变换性后，得到了暗孤子情况的平方Jost解的完备性的正确表述，并在单个暗孤子的情况利用平方Jost解的显式直接验证了它的正确性。

All equilibrium points of the original system are stable equilibrium points in the adjoint system.

对于一个非线性自治动力系统，存在一个伴随系统，该伴随系统是一个梯度系统，原始系统的所有平衡点都是伴随系统的渐近稳定平衡点，并且每一平衡点存在解析形式的Lyapunov函数。

For a nonlinear autonomous dynamic system, there exists an adjoint system, which is a gradient system. All equilibrium points of the original system are stable equilibrium points in the adjoint system.

对于一个非线性自治动力系统，存在一个伴随系统，该伴随系统是一个梯度系统，原始系统的所有平衡点都是伴随系统的渐近稳定平衡点，并且每一平衡点存在解析形式的Lyapunov函数。

The optimal control law obtained consists of linear analytic functions and a compensation term which is a series sum of the adjoint vectors. The analytic functions can be found by solving a Riccati matrix difference equation and a matrix difference equation. The compensation term can be obtained by a recursion formula that solves adjoint vector equations.

得到的最优输出跟踪控制律由状态向量的线性解析函数和伴随向量级数形式的补偿项组成，其解析函数由一次性求解Riccati矩阵差分方程和矩阵差分方程得到，补偿项由求解伴随向量差分方程的递推公式得到。

Let Un the Chebyshev polynomials of the second kind and the associated functions In this paper we discuss the approximations of the partial sums for the biorthogonal series based on and the corresponding conjugate series.

记Un是第二类Chebyshev多项式，伴随函数，这里讨论基于的双正交级数和其共轭级数的部分和逼近问题。

The method is firstly applied to solve the linear convection-dominated diffusion problems and the optimal L2-error estimates of the unknown function c and the vector flux p are gained.

此方法首先被用于求解线性对流扩散问题，得到了关于未知函数c及伴随函数p的最优L~2误差估计。

I didn't watch TV last night, because it .

昨晚我没有看电视，因为电视机坏了。

Since this year, in a lot of villages of Beijing, TV of elevator liquid crystal was removed.

今年以来，在北京的很多小区里，电梯液晶电视被撤了下来。