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An affine control system has the form (i) $\\dot x = X_0 (x) + \\sum\\nolimits_{i - 1}^m {u_i } X_i (x)$; a second-order partial differential equation can be written as (ii) $\\sum\\nolimits_{i = 1}^m {X_i^2 } v - X_0 v = f$, while if $X_0 (p) = 0$ the asymptotic stability of the rest solution requires analysis of (iii) $\\dot x = X_0 (x)$. If, in (ii), $m &lt; n$ yet the vector fields in the Lie algebra generated by $X_1 , \\cdots ,X_m $ , when evaluated at p, span the tangent space to M at p, the operator is hypoelliptic but an approximation of the vector fields needed to describe the singularity in a parametrix must retain more information than a linearization of the vector fields does. Similarly, the question of small time local controllability of (i) has been dealt with by constructing higher-order approximating vector fields which generate a nilpotent Lie algebra. The theory of these high-order approximations and their applications is concisely developed.<\/jats:p>","DOI":"10.1137\/1033050","type":"journal-article","created":{"date-parts":[[2005,3,7]],"date-time":"2005-03-07T02:21:47Z","timestamp":1110162107000},"page":"238-264","source":"Crossref","is-referenced-by-count":284,"title":["Nilpotent and High-Order Approximations of Vector Field Systems"],"prefix":"10.1137","volume":"33","author":[{"given":"Henry","family":"Hermes","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,18]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0328050"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6911(89)90100-X"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1080\/0020718508961201"},{"key":"R4","series-title":"Progr. 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