School of Mathematics and Statistics, Wuhan
2 School of Mathematical Sciences, Fudan University, KLMNS, Shanghai 200433, China
Corresponding author: firstname.lastname@example.org
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.
Mathematics Subject Classification: 34H15 / 49N20
Key words: Linear time-periodic controlled ODEs / periodic stabilization / null-controllable subspaces / the transformation over time T
This author was partially supported by the National Natural Science Foundation of China under grants 11161130003 and 11171264 and by the National Basis Research Program of China (973 Program) under grant 2011CB808002.
© EDP Sciences, SMAI, 2014