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Periodic stabilization for linear time-periodic ordinarydifferential equations∗∗∗

Published online by Cambridge University Press:  27 January 2014

Gengsheng Wang and
Yashan Xu
Gengsheng Wang
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China. wanggs62@yeah.net
Yashan Xu
Affiliation:
School of Mathematical Sciences, Fudan University, KLMNS, Shanghai 200433, China; Corresponding author: yashanxu@fudan.edu.cn

Abstract

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This paper studies the periodic feedback stabilization of the controlled lineartime-periodic ordinary differential equation:ẏ(t) = A(t)y(t) + B(t)u(t),t ≥ 0, where [A(·), B(·)] is aT-periodic pair, i.e.,A(·) ∈ L∞(ℝ+;ℝn×n) andB(·) ∈ L∞(ℝ+;ℝn×m) satisfy respectivelyA(t + T) = A(t)for a.e. t ≥ 0 andB(t + T) = B(t)for a.e. t ≥ 0. Two periodic stablization criteria for aT-period pair [A(·), B(·)] areestablished. One is an analytic criterion which is related to the transformation over timeT associated with A(·); while another is a geometriccriterion which is connected with the null-controllable subspace of[A(·), B(·)]. Two kinds of periodic feedback lawsfor a T-periodically stabilizable pair [ A(·),B(·) ] are constructed. They are accordingly connected with two Cauchy problemsof linear ordinary differential equations. Besides, with the aid of the geometriccriterion, we find a way to determine, for a given T-periodicA(·), the minimal column number m, as well as atime-invariant n×m matrix B, suchthat the pair [A(·), B] isT-periodically stabilizable.

Keywords

Linear time-periodic controlled ODEsperiodic stabilizationnull-controllable subspacesthe transformation over time T

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014, pp. 269 - 314
Copyright
© EDP Sciences, SMAI, 2014

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