本文進一步用分析法求解了一般的矩陣Riccati微分方程。線性系統兩端邊值問題與Riccati微分方程是緊密相關的。對于m×n的解矩陣，相關的線性系統為n維與m維的一對微分方程，其邊界條件在兩端0t=t與ft=t處分別為給定n維向量與m維向量?；谙嚓P線性系統的系統矩陣的本征解，提供了向前與向后積分的兩個Riccati微分方程分析解。通常的Riccati微分方程，對于線性二次（LQ）最優控制或對于Kalman-Buc濾波，皆為當前方程的特殊情況。數例表明當前方法得到的數值結果是高度精確的，分析解與精細積分所得結果有十位數字相一致。將解代回代數Riccati方程，有12位以上滿足方程。

對于哈密頓體系的偏微分方程分離變量，導致哈密頓型微分方程及本征值問題。再導向哈密頓核的積分方程。證明其本征向量的共軛辛正交歸一關系。采用層疊核與積分方程泛函取最大值的變分原理，給出辛本征向量展開式，并證明其完備性定理。

本文將電磁波導的基本方程導向了Hamilton體系、辛幾何的形式。辛體系可以用于任意的各向異性材料，而且便于處理不同介質的界面條件。橫向的電場和磁場構成了對偶向量。分離變量，Hamilton算子矩陣本征值問題，共軛辛正交歸一關系，本征解的展開定理等整套理論，適用于各種波導的課題，有利于不同截面的波導連接、與共振腔的連接等，這提供了很大方便。辛體系在應用力學中的應用已經取得了很大成功，對于本征解的求解已經發展了許多方法，不同學科之間的交錯對于電磁波導的分析是很有利的。

進一步完善板彎曲與平面彈性問題的多類變量變分原理，給出了相關邊界積分項的具體表達式。多類變量變分原理涵蓋了平衡、應力函數—應力、位移—應變、協調和物性共五大類基本方程和所有邊界條件，是一個具有更加廣泛意義的變分原理。

本文建立平面彈性與板彎曲的相似性理論，給出了板彎曲經典理論的另一套基本方程與求解方法，然后進入哈密頓體系用直接法研究板彎曲問題。新方法論應用分離變量、本征函數展開方法給出了條形板問題的分析解，突破了傳統半逆解法的限制。結果表明新方法論有廣闊的應用前景。

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

根據結約力學與最優控制的模擬關系，H∞狀態反饋控制的最優參數Y-2a與柱形結構穩定性分析或振動中的Rayleigh商一階本征值相對應，這是Ya是閉環系統的最優H∞范數。在大系統控制中，分散控制方法通過將整個系統劃分成一系列子系統來研究系統的控制問題，這與結構力學中的子結構分析技術類似。本文由此建立了H∞分散控制與子結構振動問題之間的對應關系，指出大系統分散控制的最優參數Y-2a對應于整體結構的一階本征值，并利用模態綜合法計算H∞分散控制問題中的最優參數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.