# Periodic stabilization for linear time-periodic ordinary differential equations

ESAIM: Control, Optimisation and Calculus of Variations (2014)

- Volume: 20, Issue: 1, page 269-314
- ISSN: 1292-8119

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topWang, Gengsheng, and Xu, Yashan. "Periodic stabilization for linear time-periodic ordinary differential equations." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 269-314. <http://eudml.org/doc/272841>.

@article{Wang2014,

abstract = {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.},

author = {Wang, Gengsheng, Xu, Yashan},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {linear time-periodic controlled odes; periodic stabilization; null-controllable subspaces; the transformation over time T; linear time-periodic controlled ODEs; the transformation over time $T$},

language = {eng},

number = {1},

pages = {269-314},

publisher = {EDP-Sciences},

title = {Periodic stabilization for linear time-periodic ordinary differential equations},

url = {http://eudml.org/doc/272841},

volume = {20},

year = {2014},

}

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

PB - EDP-Sciences

VL - 20

IS - 1

SP - 269

EP - 314

AB - 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.

LA - eng

KW - linear time-periodic controlled odes; periodic stabilization; null-controllable subspaces; the transformation over time T; linear time-periodic controlled ODEs; the transformation over time $T$

UR - http://eudml.org/doc/272841

ER -

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