Periodic stabilization for linear time-periodic ordinary differential equations

Gengsheng Wang; Yashan Xu

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 1, page 269-314
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Wang, Gengsheng, and Xu, Yashan. "Periodic stabilization for linear time-periodic ordinary differential equations." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 269-314. <http://eudml.org/doc/272841>.

@article{Wang2014,
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.},
author = {Wang, Gengsheng, Xu, Yashan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {linear time-periodic controlled odes; periodic stabilization; null-controllable subspaces; the transformation over time T; linear time-periodic controlled ODEs; the transformation over time $T$},
language = {eng},
number = {1},
pages = {269-314},
publisher = {EDP-Sciences},
title = {Periodic stabilization for linear time-periodic ordinary differential equations},
url = {http://eudml.org/doc/272841},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Wang, Gengsheng
AU - Xu, Yashan
TI - Periodic stabilization for linear time-periodic ordinary differential equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 1
SP - 269
EP - 314
AB - 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.
LA - eng
KW - linear time-periodic controlled odes; periodic stabilization; null-controllable subspaces; the transformation over time T; linear time-periodic controlled ODEs; the transformation over time $T$
UR - http://eudml.org/doc/272841
ER -

References

top
  1. [1] 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). Zbl1027.93001MR1997753
  2. [2] 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). Zbl0648.34002MR947141
  3. [3] V.I. Arnol′d, Ordinary Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, Budapest (2003). Zbl0744.34001
  4. [4] V. Barbu and G. Wang, Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems. Indiana Univ. Math. J.54 (2005) 1521–1546. Zbl1176.93060MR2189677
  5. [5] R. Brockett, A stabilization problem, Open problems in mathematical systems and control theory. Commun. Control Engrg. Ser. Springer, London (1999) 75–78. MR1727940
  6. [6] P. Brunovský, Controllability and linear closed-loop controls in linear periodic systems. J. Differ. Equs.6 (1969) 296–313. Zbl0176.06301MR243180
  7. [7] J.M. Coron, Control and nonlinearity, Mathematical Surveys and Monographs, vol. 136 of Amer. Math. Soc. Providence, RI (2007). Zbl1140.93002MR2302744
  8. [8] G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques (in French). Ann. Sci. École Norm. Sup. 12 (1883) 47–88. JFM15.0279.01
  9. [9] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lect. Notes Math. Springer-Verlag, Berlin, New York (1981). Zbl0456.35001MR610244
  10. [10] 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). Zbl0309.34001MR486784
  11. [11] M. Ikeda, H. Maeda and S. Kodama, Stabilization of linear systems. SIAM J. Control10 (1972) 716–729. Zbl0244.93049MR335015
  12. [12] M. Ikeda, H. Maeda and S. Kodama, Estimation and feedback in linear time-varying systems: a deterministic theory. SIAM J. Control13 (1975) 304–326. Zbl0305.93037MR429190
  13. [13] H. Kano and T. Nishimura, Periodic solution of matrix Riccati equations with detectability and stabilizability. Internat. J. Control29 (1979) 471–487. Zbl0409.93037MR529183
  14. [14] H. Kano and T. Nishimura, Controllability, stabilizability, and matrix Riccati equations for periodic systems. IEEE Trans. Automat. Control30 (1985) 1129–1131. Zbl0577.93004MR810315
  15. [15] G.A. Leonov, The Brockett stabilization problem (in Russian). Avtomat. i Telemekh (2001) 190–193; translation in Autom. Remote Control 62 (2001) 847–849. Zbl1092.93589MR1852759
  16. [16] X. Li, J. Yong and Y. Zhou, Control Theory (in Chinese). Higher Education Press of P.R. China, Beijing (2009). 
  17. [17] A.M. Lyapunov, 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. Zbl0786.70001MR1154209
  18. [18] I.G. Malkin, The stability theory of motion (in Russian). Nauk Press, Moscow (1966). Zbl0136.08502MR206419
  19. [19] R. Penrose, A Generalized inverse for matrices. Proc. Cambridge Philos. Soc.51 (1955) 406–413. Zbl0065.24603MR69793
  20. [20] E.D. Sontag, Mathematical control theory: Deterministic finite-dimensional systems, 2nd edition, vol. 6 of Texts in Applied Mathematics. Springer-Verlag, New York (1998). Zbl0945.93001MR1640001
  21. [21] 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). Zbl1239.15017MR2807612
  22. [22] 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). Zbl0991.34001MR1629775
  23. [23] K. Yosida, Functional analysis, 6th edition, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, New York (1980). Zbl0435.46002MR617913

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.