A Category of Control Systems

Abstract

We construct the concrete category LiCS of left-invariant control systems (on Lie groups) and point out some very basic properties. Morphisms in this category are examined briefly. Also, covering control systems are introduced and organized into a (comma) category associated with LiCS

Two-input control systems on the Euclidean group SE (2)

“…The trace Γ = im Ξ (1, •) is a submanifold of g so that Γ = Ξ u = Ξ (1, u) : u ∈ R (cf. [5,6]). A left-invariant control affine system is one whose parametrisation map is affine.…”

Control affine systems on solvable three-dimensional Lie groups, I

“…We shall denote such a system by Σ = (G, Ξ) (cf. [3]). The admissible controls are piecewise continuous maps u(·) : [0, T ] → R .…”

“…We specialize feedback equivalence (in the context of left-invariant control systems) by requiring that the feedback transformations are left-invariant (i.e., constant over the state space). Such transformations are exactly those that are compatible with the Lie group structure (see, e.g., [3,2]). More precisely, let Σ = (G, Ξ) and Σ = (G , Ξ ) be left-invariant control affine systems.…”

“…In this paper we classify, under L-equivalence, the (full-rank) affine subspaces of g 3.1 , g 3.2 , and g 3. 3 . Clearly, if Γ 1 and Γ 2 are L-equivalent, then they are necessarily of the same dimension.…”

Equivalence of Control Systems on the Pseudo-Orthogonal Group SO (2, 1)0

“…Such systems were first considered in 1972 by Brockett [12] and by Jurdjevic and Sussmann [17]. For more details about (invariant) control systems see, e.g., [5], [16], [24], [6], [22].…”