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

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⁺; 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.