Grigory Sklyar
University of Szczecin, Poland
Representation of affine control systems as series of nonlinear power moments and optimality
We develop the approach to the time-optimal problem based on the Markov moment problem method. The initial point in
our analysis is the representation of affine control systems in a neighborhood of the equilibrium as the series of
nonlinear power moments. That allows, in particular, to examine analytic changes of variables in the system as the
corresponding analytic transformations of its series. We consider the free algebra of nonlinear power moments A an
d study the properties of finite-generated right ideals of this algebra. In this way we obtain the following genera
lization of the remarkable R. Ree's theorem:
Let J be an ideal generated by finite number of Lee elements of A and J^\perp be its orthogonal complement in A. Th
en J^\perp admits the orthogonal decomposition
J^\perp = L \oplus L^{sh},
where L is the orthogonal projection of the Lee subalgebra of A to J^\perp and
L^{sh} = \{l1* l2* . . . *lk, k \geq2, l_j\in L\} , where * means the shuffle product in A.
Using this theorem we obtain the result on the reduction (by an analytic transformation) of the system series to th
e canonical form and define the (homogeneous) principle part of the series. This, in turn, gives the asymptotic app
roximation of the time optimal problems for affine systems by problems for the homogeneous systems whose series hav
e the same principle part.
Joint work with Svetlana Ignatovich (Kharkov National University).
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