Communications in Information and Systems

Volume 23 (2023)

Number 1

From dimension-free manifolds to dimension-varying control systems

Pages: 85 – 150



Daizhan Cheng (Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, C.A.S., Beijing, China; and Research Center of Semi-tensor Product of Matrices, Theory and Applications, Liaocheng University, Lianocheng, China)

Zhengping Ji (Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, C.A.S., Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)


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 dimensionvarying 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.


cross-dimensional projection, dimensionvarying dynamic (control) systems, dimension-free Euclidean spaces (manifolds), dimension-free dynamic (control) systems

2010 Mathematics Subject Classification

34H05, 37N35, 51H25

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The authors were supported partly by the National Natural Science Foundation of China (NSFC) under Grant 62073315.

Received 15 February 2023

Published 17 April 2023