## Periodic stabilization for linear time-periodic ordinary
differential equations^{∗,}^{∗∗}

^{1}
School of Mathematics and Statistics, Wuhan
University, Wuhan,
430072,
China

wanggs62@yeah.net

^{2}
School of Mathematical Sciences, Fudan University, KLMNS,
Shanghai
200433,
China

Corresponding author: yashanxu@fudan.edu.cn

Received:
23
October
2012

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.

Mathematics Subject Classification: 34H15 / 49N20

Key words: Linear time-periodic controlled ODEs / periodic stabilization / null-controllable subspaces / the transformation over time *T*

*© EDP Sciences, SMAI, 2014*