论文标题
从无维的流形到尺寸变化的控制系统
From Dimension-Free Manifolds to Dimension-varying Control Systems
论文作者
论文摘要
提出了从矢量乘数，内部产物，规范，距离以及添加不同维度的两个向量的添加，这使得将空间变成拓扑矢量空间，称为欧几里得空间不同维度（ESDD）。通过距离获得等效性。作为ESDDS W.R.T.的商空间获得了等效性，无尺寸的欧几里得空间（DFESS）和无尺寸的歧管（DFM），它们在每个点处都将其捆绑为圆形矢量空间作为其切线空间。使用从ESDD到DFE的自然投影，获得了纤维束结构，该结构的总空间为总空间，DFE作为其基本空间。鉴别几何形状（例如平滑函数，（共同）矢量场，张量场等）中的经典对象已借助不同维欧几里得空间之间的投影扩展到DFM的情况。然后呈现了尺寸变化的动态系统（DVDS）和维度变化控制系统（DVCSS），它们以DFM作为状态空间。实现是将DVDS或DVCS从DFM中提升为ESDD的，并且研究了DVDS或DVCS从ESDDS的投影到DFMS上。
Starting from the vector multipliers, the inner product, norm, distance, as well as addition of two vectors of different dimensions are proposed, which makes the spaces into a topological vector space, called the Euclidean space of different dimension (ESDD). An equivalence is obtained via distance. As a quotient space of ESDDs w.r.t. equivalence, the dimension-free Euclidean spaces (DFESs) and dimension-free manifolds (DFMs) are obtained, which have bundled vector spaces as its tangent space at each point. Using the natural projection from a ESDD to a DFES, a fiber bundle structure is obtained, which has ESDD as its total space and DFES as its base space. Classical objects in differential geometry, such as smooth functions, (co-)vector fields, tensor fields, etc., have been extended to the case of DFMs with the help of projections among different dimensional Euclidean spaces. Then the dimension-varying dynamic systems (DVDSs) and dimension-varying control systems (DVCSs) are presented, which have DFM as their state space. The realization, which is a lifting of DVDSs or DVCSs from DFMs into ESDDs, and the projection of DVDSs or DVCSs from ESDDs onto DFMs are investigated.
