On the relation between small-time local controllability and normal self-reachability

On the existence of stabilising feedback controls for real analytic small-time locally controllable systems

“…The control system f is said to have the non-tangency property at a point p ∈ P if it is not tangent to any immersed submanifold of M that contains p and has positive codimension. If f is STLC from a point p ∈ M, 6 then Theorem 4.14 of [27] implies that f has the non-tangency property at p and, hence, by Theorem 5.3 of the same reference, the time-reversed system − f is also STLC from p (this is important for the proof of Theorem 2, below, because it is necessary to consider trajectories of the time-reversed system − f ). Moreover, by the same Theorem 5.3, the point p is small-time normally self-reachable via both systems f and − f .…”

“…This apparent impasse can be overcome by invoking the property of normal reachability; the relevant argument is explained in the next paragraph. Unless stated otherwise, the reference for the following paragraph is [27].…”

“…If we let U meas denote the set of measurable functions u : I ⊂ R → Ω, where I is an interval, then the class of admissible controls for a control system f is defined to be the subset U f ⊂ U meas containing the functions u with the property that the differential equationẋ(t) = f (x(t), u(t)) satisfies Carathéodory's theorem on the existence and uniqueness of solutions. If T f ω : TM × Ω → TTM is continuous 3 -a property we always assume-then U f contains the set U step of piecewise constant maps with a finite number of discontinuities [27,28] Because the objects of interest in the paper are real analytic control systems, in the remainder of the paper, and without loss of generality [27], the admissible controls will be elements of U step . This choice is implicit in the definitions of Sect.…”

“…To be able to construct such a feedback, not only need we know that each point in a reachable set of an STLC system can be reached using a piecewise constant control, but also that the concatenation of flows that corresponds to the piecewise constant control has rank equal to the dimension of the state space, as a map from the space of switching times to the state space of the control system. This last property is called normal reachability; the formal definition is as follows [27].…”

“…To avoid cumbersome expressions, we sometimes write Γ ω E for the space of real analytic sections of a vector bundle (E, π, M). We begin by making precise what is meant by a "control system"; the definition is adopted from [27,28,47].…”