Ivancevic_Applied-Diff-Geom

538 Applied Differential Geometry: A Modern Introductionu. For a fixed U we denote by U the collection of all measurable controlstaking their values in U. To be concise about this, a control affine systemis a triple Σ = (M, F = {f 0 , f 1 , ..., f m }, U), with all objects as definedabove. A controlled trajectory for Σ is a pair (c, u), where u ∈ U and wherec : [0, τ(u)] → M is defined so thatċ(t) = f 0 (c(t)) + u i (t)f i (c(t)).One can show that for admissible controls, the curve c will exist at least forsufficiently small times, and that the initial condition c(0) = x 0 uniquelydefines c on its domain of definition.For x ∈ M and T > 0 we define several types of reachable sets as:R Σ (x, T ) = {c(T ) :(c, u) is a controlled trajectory for Σ with τ(u) = T and c(0) = x},R Σ (x, ≤ T ) = ∪ t∈[0,T ] R Σ (x, t),R Σ (x) = ∪ t≥0 R Σ (x, t),that allow us to give several definitions of controllability as follows. LetΣ = (M, F, U) be a control affine system and let x ∈ M. We say that:(1) Σ is accessible from x if int(R Σ (x)) ≠ 0.(2) Σ is strongly accessible from x if int(R Σ (x, T )) ≠ 0 for each T > 0.(3) Σ is locally controllable from x if x ∈ int(R Σ (x)).(4) Σ is small–time locally controllable (STLC) from x if there exists T > 0so that x ∈ int(R Σ (x, ≤ T )) for each t ∈ [0, T ].(5) Σ is globally controllable from x if (R Σ (x)) = M.For example, a typical simple system that is accessible but not controllableis given by the following data:M = R 2 , m = 1, U = [−1, 1],ẋ = u, ẏ = x 2 .This system is (not obviously) accessible from (0, 0), but is (obviously) notlocally controllable from that same point. Note that although R Σ ((0, 0), ≤T ) has nonempty interior, the initial point (0, 0) is not in that interior.Thus this is a system that is not controllable in any sense. Note that thesystem is also strongly accessible.