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

  • Andrei A. Agrachev
Part of the Communications and Control Engineering book series (CCE)


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 xR 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 = f ij (x), for some i j ∈ {1,…,k}, such that ξ0(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.


Vector Field Local Controllability Steklov Mathematical Institute Taylor Polynomial Scalar Input 
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© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Andrei A. Agrachev
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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