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Communications in Information and Systems
Volume 23 (2023)
From dimension-free manifolds to dimension-varying control systems
Pages: 85 – 150
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
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