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^{+}; ℝ^{n×n}) and B(·) ∈ L^{∞}(ℝ^{+}; ℝ^{n×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://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/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, Volume 20 (2014) no. 1, pp. 269-314. doi : 10.1051/cocv/2013064. http://archive.numdam.org/articles/10.1051/cocv/2013064/

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