区段混合能的偏微分方程

结构力学与最优控制模拟关系的共同基础就是分析力学，表明在结构力学与最优控制理论的架构内也应有分析力学的整套理论。传统分析力学总是考虑连续时间、同时的状态、不变维数的体系。并且物性为即时响应的。但结构力学有限元要考虑离散坐标、不同坐标状态、而且变动维数、时滞的体系。根据区段变形能只与其两端位移有关，就可通过数学分析得到Lagrange括号与Poisson括号等内容。区段变形能就是作用量，满足Hamilton-Jacobi方程。但还有区段混合能的表示，本文证明它满足雷同的偏微分方程。它们在离散体系时还有偏差分方程。本文进一步给出了其与Riccati方程的关系。

The PDEs for Mixed Energy Representation

The analogy theory between structural mechanics and optimal control is based on analytical mechanics, which implies that there are whole set of analytical theories in both sides of structural mechanics and optimal control. The traditional analytical dynamics treats problems with continuous-time, states at the same time, systems with same dimensions and also that the constitutive relation responded instantly. However, structural mechanics and FEM want to consider discrete coordinate, and the nodal states at nonunified coordinates, with different dimensions and some time with time-delay constitutive relations. Based on the statement that the interval deformation energy relates only on the displacements at two ends, the Lagrange bracket and Poisson bracket can be derived from pure mathematical analysis. The interval deformation energy in structural mechanics corresponds to the action function in analytical dynamics and sat- isfying the Hamilton-Jacobi equation. There is also the mixed energy representation of interval energy in structural mechanics, which is shown satisfying a similar partial differential equation in this paper. Both PDEs of two interval energy form hold corresponding partial difference equations. Furthermore, the relationships with the Riccati equations are also shown in the paper.