# Is it possible to recognize local controllability in a finite number of differentiations?

## Abstract

Let *f* _{1},…, *f* _{k}, *k* ≥ 2, be real-analytic vector fields defined on a neighborhood of the origin in *R* ^{ n }, and *t* > 0. The point *x* ∈ *R* ^{ n } is called attainable from the origin for a time less than *t* and with no more than *N* switchings, if there exists a subdivision 0 = to < *t* _{1} < … < *T* _{ N } _{+1} < *t* of the segment [0,*t*] and solutions ξ_{
j } (*t*), *t* ∈ [*t* _{ j }, *t* _{ j } _{+1}] of the differential equations *ẋ* = *f* _{ ij }(*x*), for some *i* _{ j } ∈ {1,…,*k*}, such that ξ_{0}(0) = 0, ξ _{
j } _{−1}(*t* _{ j }) = ξ_{
j }(*t* _{ j }) for *j* = 1,…, *N*, ξ_{
N }(*t* _{ N } _{+1}) = *x*. Let *A* _{ t }(*N*) be the set of all such points *x*; the set \(
{A_t} = \bigcup\limits_{N > 0} {At} (N)
\) is called the attainable set for a time no greater than *t*.

## Keywords

Vector Field Local Controllability Steklov Mathematical Institute Taylor Polynomial Scalar Input## Preview

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