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This paper studies the periodic feedback stabilization of the controlled linear time-periodic ordinary differential equation: ẏ(t) = A(t)y(t) + B(t)u(t), t ≥ 0, where [A(·), B(·)] is a T-periodic pair, i.e., A(·) ∈ L∞(ℝ+; ℝn×n) and B(·) ∈ L∞(ℝ+; ℝn×m) satisfy respectively A(t + T) = A(t) for a.e. t ≥ 0 and B(t + T) = B(t) for a.e. t ≥ 0. Two periodic stablization criteria for a T-period pair [A(·), B(·)] are established. One is an analytic criterion which is related to the transformation over time T associated with A(·); while another is a geometric criterion which is connected with the null-controllable subspace of [A(·), B(·)]. Two kinds of periodic feedback laws for a T-periodically stabilizable pair [ A(·), B(·) ] are constructed. They are accordingly connected with two Cauchy problems of linear ordinary differential equations. Besides, with the aid of the geometric criterion, we find a way to determine, for a given T-periodic A(·), the minimal column number m, as well as a time-invariant n×m matrix B, such that the pair [A(·), B] is T-periodically stabilizable.
Keywords: linear time-periodic controlled odes, periodic stabilization, null-controllable subspaces, the transformation over time T
@article{COCV_2014__20_1_269_0,
author = {Wang, Gengsheng and Xu, Yashan},
title = {Periodic stabilization for linear time-periodic ordinary differential equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {269--314},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013064},
mrnumber = {3182700},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013064/}
}
TY - JOUR AU - Wang, Gengsheng AU - Xu, Yashan TI - Periodic stabilization for linear time-periodic ordinary differential equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 269 EP - 314 VL - 20 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013064/ DO - 10.1051/cocv/2013064 LA - en ID - COCV_2014__20_1_269_0 ER -
%0 Journal Article %A Wang, Gengsheng %A Xu, Yashan %T Periodic stabilization for linear time-periodic ordinary differential equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 269-314 %V 20 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013064/ %R 10.1051/cocv/2013064 %G en %F COCV_2014__20_1_269_0
Wang, Gengsheng; Xu, Yashan. Periodic stabilization for linear time-periodic ordinary differential equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 269-314. doi: 10.1051/cocv/2013064
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