From Dimension-Free Manifolds to Dimension-varying Control Systems

TL;DR

This paper develops a mathematical framework for dimension-free Euclidean spaces and manifolds, enabling the analysis and control of systems with varying dimensions using differential geometry concepts.

Contribution

It introduces the concept of dimension-free manifolds and extends differential geometry tools to analyze dimension-varying control systems.

Findings

Defined Euclidean spaces of different dimensions (ESDD) and their quotient spaces.

Extended classical differential geometry objects to dimension-free manifolds.

Presented a framework for analyzing and controlling systems with changing dimensions.

Abstract

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.