本文进一步用分析法求解了一般的矩阵Riccati微分方程。线性系统两端边值问题与Riccati微分方程是紧密相关的。对于m×n的解矩阵，相关的线性系统为n维与m维的一对微分方程，其边界条件在两端0t=t与ft=t处分别为给定n维向量与m维向量。基于相关线性系统的系统矩阵的本征解，提供了向前与向后积分的两个Riccati微分方程分析解。通常的Riccati微分方程，对于线性二次（LQ）最优控制或对于Kalman-Buc滤波，皆为当前方程的特殊情况。数例表明当前方法得到的数值结果是高度精确的，分析解与精细积分所得结果有十位数字相一致。将解代回代数Riccati方程，有12位以上满足方程。

对于哈密顿体系的偏微分方程分离变量，导致哈密顿型微分方程及本征值问题。再导向哈密顿核的积分方程。证明其本征向量的共轭辛正交归一关系。采用层叠核与积分方程泛函取最大值的变分原理，给出辛本征向量展开式，并证明其完备性定理。

本文将电磁波导的基本方程导向了Hamilton体系、辛几何的形式。辛体系可以用于任意的各向异性材料，而且便于处理不同介质的界面条件。横向的电场和磁场构成了对偶向量。分离变量，Hamilton算子矩阵本征值问题，共轭辛正交归一关系，本征解的展开定理等整套理论，适用于各种波导的课题，有利于不同截面的波导连接、与共振腔的连接等，这提供了很大方便。辛体系在应用力学中的应用已经取得了很大成功，对于本征解的求解已经发展了许多方法，不同学科之间的交错对于电磁波导的分析是很有利的。

进一步完善板弯曲与平面弹性问题的多类变量变分原理，给出了相关边界积分项的具体表达式。多类变量变分原理涵盖了平衡、应力函数—应力、位移—应变、协调和物性共五大类基本方程和所有边界条件，是一个具有更加广泛意义的变分原理。

本文建立平面弹性与板弯曲的相似性理论，给出了板弯曲经典理论的另一套基本方程与求解方法，然后进入哈密顿体系用直接法研究板弯曲问题。新方法论应用分离变量、本征函数展开方法给出了条形板问题的分析解，突破了传统半逆解法的限制。结果表明新方法论有广阔的应用前景。

An analytical criterion of patch test is proposed, which is proved to be equivalent to the engineering version, and also proved to be the necessary and sufficient condition of FEM convergence, the weak variational crime condition is also explained.

When using techniques to design de-centralized controllers for large systems, the whole system is divided into subsystems which are analysed using control theory before being re-combined. An analogy is established with substructural analysis in structural mechanics, in which ∞H∞H∞H de-centralized control theory corresponds to substructural modal synthesis theory so that the optimal parameter 2crγ− of the whole system corresponds to the fundamental vibration frequency of the whole structure, here crγ is the optimal ∞H induced norm of the whole system, and its determination is a key step in the design of ∞H de-centralized control systems. Hence modal synthesis methodology and the extended Wittrick-Williams

根据结约力学与最优控制的模拟关系，H∞状态反馈控制的最优参数Y-2a与柱形结构稳定性分析或振动中的Rayleigh商一阶本征值相对应，这是Ya是闭环系统的最优H∞范数。在大系统控制中，分散控制方法通过将整个系统划分成一系列子系统来研究系统的控制问题，这与结构力学中的子结构分析技术类似。本文由此建立了H∞分散控制与子结构振动问题之间的对应关系，指出大系统分散控制的最优参数Y-2a对应于整体结构的一阶本征值，并利用模态综合法计算H∞分散控制问题中的最优参数Y-2a。

The Hamiltonian system method is applied to electro-magnetic wave-guides to permit symplectic analysis to be applied to arbitrary anisotropic materials and to permit the interface conditions between segments of a periodical wave-guide to be treated. The transverse electric and magnetic fields compose the dual vectors. An electro-magnetic stiffness matrix is introduced which relates to the electric field vectors at both its ends. The pass-and stop-band solutions are given for a constant cross-section segment of a plane wave-guide and then the equations neede to combine two segments with different cross-magnetic stiffness matrix of a fundamental period of the wave-guide, which is computed by combining segments recursively until the shortest repeating length of the wave-guide has been assembled. Finally, the pass-band eigenvalues of the energy band are found by using the Wittrick-Williams algorithm throughout the calculations, starting with the combination of segments step.

Under the computer impact, optimal control theory is proposed based on the state space model, which should at least trace back to the Hamilton canonical equation system. On the other hand, introducing the state space approach, the dual equations derived in structural mechanics have the same form as those in optimal control theory, and then the analogy between structural mechanics and optimal control is established. Both the LQ control and the Kalman-Bucy filter correspond to structural static, and both the H control and H filter critical (optimal) parameters correspond to Euler critical load factor in structural stability, i.e. the extended Rayleigh quotient. Further, the parameter identification of plant matrix corresponds to inverse problem in structural analysis, and also the large scale de-centralized control corresponds to sub-structural combination and its H critical parameter can be solved by the modal synthesis method in structural vibrations. All these relationships of analogy theory and the respective precise integration method are described in this paper, perspectively.