Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-12T19:41:17.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic stabilization for linear time-periodic ordinary differential 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. [email protected]
Yashan Xu
Affiliation:
School of Mathematical Sciences, Fudan University, KLMNS, Shanghai 200433, China; Corresponding author: [email protected]
Get access

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.

Keywords

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati equations: In control and systems theory. Systems & Control: Foundations & Applications. Birkhäuser Verlag, Basel (2003).
V.I. Arnol′d, Geometrical methods in the theory of ordinary differential equations. Translated from the Russian by Joseph Szäcs. 2nd edition, vol. 250, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York (1988).
V.I. Arnol′d, Ordinary Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, Budapest (2003).
Barbu, V. and Wang, G., Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems. Indiana Univ. Math. J. 54 (2005) 1521–1546. Google Scholar
R. Brockett, A stabilization problem, Open problems in mathematical systems and control theory. Commun. Control Engrg. Ser. Springer, London (1999) 75–78.
Brunovský, P., Controllability and linear closed-loop controls in linear periodic systems. J. Differ. Equs. 6 (1969) 296–313. Google Scholar
J.M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs, vol. 136 of Amer. Math. Soc. Providence, RI (2007).
Floquet, G., Sur les équations différentielles linéaires à coefficients périodiques (in French). Ann. Sci. École Norm. Sup. 12 (1883) 47–88. Google Scholar
D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lect. Notes Math. Springer-Verlag, Berlin, New York (1981).
M.W. Hirsch and S. Smale, Differential equations, dynamical systems, and linear algebra, vol. 60 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London (1974).
Ikeda, M., Maeda, H. and Kodama, S., Stabilization of linear systems. SIAM J. Control 10 (1972) 716–729. Google Scholar
Ikeda, M., Maeda, H. and Kodama, S., Estimation and feedback in linear time-varying systems: a deterministic theory. SIAM J. Control 13 (1975) 304–326. Google Scholar
Kano, H. and Nishimura, T., Periodic solution of matrix Riccati equations with detectability and stabilizability. Internat. J. Control 29 (1979) 471–487. Google Scholar
Kano, H. and Nishimura, T., Controllability, stabilizability, and matrix Riccati equations for periodic systems. IEEE Trans. Automat. Control 30 (1985) 1129–1131. Google Scholar
Leonov, G.A., The Brockett stabilization problem (in Russian). Avtomat. i Telemekh (2001) 190–193; translation in Autom. Remote Control 62 (2001) 847–849. Google Scholar
X. Li, J. Yong and Y. Zhou, Control Theory (in Chinese). Higher Education Press of P.R. China, Beijing (2009).
Lyapunov, A.M., The general problem of the stability of motion, Translated from Edouard Davaux’s French translation (1907) of the 1892 Russian original and edited by A.T. Fuller. With an introduction and preface by Fuller, a biography of Lyapunov by V.I. Smirnov, and a bibliography of Lyapunov’s works compiled by J.F. Barrett. Lyapunov centenary issue. Reprint of Internat. J. Control 55 (1992) 521–790. Google Scholar
I.G. Malkin, The stability theory of motion (in Russian). Nauk Press, Moscow (1966).
Penrose, R., A Generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51 (1955) 406–413. Google Scholar
E.D. Sontag, Mathematical control theory: Deterministic finite-dimensional systems, 2nd edition, vol. 6 of Texts in Applied Mathematics. Springer-Verlag, New York (1998).
W.H. Steeb and Y. Hardy, Matrix calculus and Kronecker product, A practical approach to linear and multilinear algebra, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011).
W. Walter, Ordinary differential equations, Translated from the sixth German (1996) edition by Russell Thompson, vol. 182 of Graduate Texts in Mathematics Readings in Mathematics. Springer-Verlag, New York (1998).
K. Yosida, Functional analysis, 6th edition, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, New York (1980).