Abstract
This paper is the applied counterpart to previous results [5] for linear-analytic control systems. It is mainly concerned with two canonical representations of the exponential type. They exhibit the Lie algebraic structure of the system in such a form that results on weak controllability are easily derived in an algebraic manner. The first representation is a single exponential of a canonical Lie series in Hall's basis of the Lie algebra of vector fields. The second one is a factorization in terms of simpler exponentials of Hall's basic vectors. Both of them exhibit, as canonical coefficients, an infinite set of characteristic parameters which are a minimal representation of the input paths, when no drift occurs in the system (or, equivalently, in the weak control case). The weak controllability theorem is easily derived from these results, in a purely algebraic way.
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Huillet, T., Monin, A. & Salut, G. Lie algebraic canonical representations in nonlinear control systems. Math. Systems Theory 20, 193–213 (1987). https://doi.org/10.1007/BF01692065
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DOI: https://doi.org/10.1007/BF01692065