Stability of differential inclusions: A computational approach

Azhmyakov, Vadim

doi:https://doi.org/10.1155/MPE/2006/17837

Mathematical Problems in Engineering

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Volume 2006 | Article ID 017837 | https://doi.org/10.1155/MPE/2006/17837

Stability of differential inclusions: A computational approach

Vadim Azhmyakov¹

Received08 Oct 2004

Revised27 Dec 2004

Accepted20 Apr 2005

Published03 Jan 2006

Abstract

We present a technique for analysis of asymptotic stability for a class of differential inclusions. This technique is based on the Lyapunov-type theorems. The construction of the Lyapunov functions for differential inclusions is reduced to an auxiliary problem of mathematical programming, namely, to the problem of searching saddle points of a suitable function. The computational approach to the auxiliary problem contains a gradient-type algorithm for saddle-point problems. We also extend our main results to systems described by difference inclusions. The obtained numerical schemes are applied to some illustrative examples.

References

F. Amato, Robust Control of Linear Systems Subject to Time-Varying Parameters, Springer, Berlin, 2005.
K. J. Arrov, L. Hurvitz, and H. Uzawa, Studies in Linear and Nonlinear Programming, Stanford University Press, Stanford, 1958.
J.-P. Aubin and A. Cellina, Differential Inclusions, vol. 264 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 1984.
View at: Zentralblatt MATH | MathSciNet
V. Azhmyakov, “A constructive method for solving stabilization problems,” Discuss. Math. Differ. Incl. Control Optim., vol. 20, no. 1, pp. 51–62, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
B. R. Barmish, “Necessary and sufficient conditions for quadratic stabilizability of an uncertain system,” J. Optim. Theory Appl., vol. 46, no. 4, pp. 399–408, 1985.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
F. Bucci, “Absolute stability of feedback systems in Hilbert spaces,” in Optimal Control (Gainesville, Fla, 1997), W. H. Hager and P. M. Pardalos, Eds., vol. 15 of Appl. Optim., pp. 24–39, Kluwer Academic, Dordrecht, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
K. Deimling, Multivalued Differential Equations, vol. 1 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, 1992.
View at: Zentralblatt MATH | MathSciNet
A. L. Dontchev, “Discrete approximations in optimal control,” in Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control (Minneapolis, Minn, 1993), B. S. Mordukhovich and H. J. Sussmann, Eds., vol. 78 of IMA Vol. Math. Appl., pp. 59–80, Springer, New York, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. L. Dontchev and F. Lempio, “Difference methods for differential inclusions: a survey,” SIAM Rev., vol. 34, no. 2, pp. 263–294, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, vol. 62 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1999.
View at: Zentralblatt MATH | MathSciNet
A. V. Fiacco and G. P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York, 1968.
View at: Zentralblatt MATH | MathSciNet
A. F. Filippov, “Stability for differential equations with discontinuous and multivalued right-hand sides,” Differentsial'nye Uravneniya, vol. 15, no. 6, pp. 1018–1027, 1148–1149, 1979 (Russian).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, vol. 18 of Mathematics and Its Applications (Soviet Series), Kluwer Academic, Dordrecht, 1988.
View at: Zentralblatt MATH | MathSciNet
É. Gyurkovics, “Receding horizon control for the stabilization of nonlinear uncertain systems described by differential inclusions,” J. Math. Systems Estim. Control, vol. 6, no. 3, pp. 1–16, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, 1966.
View at: Zentralblatt MATH | MathSciNet
D. Hinrichsen and A. J. Pritchard, “Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials,” Internat. J. Control, vol. 55, no. 4, pp. 809–844, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Johansson, Piecewise Linear Control Systems, vol. 284 of Lecture Notes in Control and Information Sciences, Springer, Berlin, 2003.
View at: Zentralblatt MATH | MathSciNet
R. E. Kalman, “Lyapunov functions for the problem of Lur'e in automatic control,” Proc. Natl. Acad. Sci. USA, vol. 49, pp. 201–205, 1963.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. A. Kamenetskiy and Ye. S. Pyatnitskiy, “An iterative method of Lyapunov function construction for differential inclusions,” Systems Control Lett., vol. 8, no. 5, pp. 445–451, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. L. Kharitonov, “Asymptotic stability of an equilibrium position of a family of systems of linear differential equations,” Differentsial'nye Uravneniya, vol. 14, pp. 1483–1485, 1979 (Russian).
View at: Google Scholar | Zentralblatt MATH
M. Kisielewicz, Differential Inclusions and Optimal Control, vol. 44 of Mathematics and Its Applications (East European Series), Kluwer Academic, Dordrecht; PWN—Polish Scientific Publishers, Warsaw, 1991.
View at: Zentralblatt MATH | MathSciNet
N. N. Krasovskiĭ, Stability of Motion. Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Stanford University Press, California, 1963.
View at: MathSciNet
A. I. Lourie and V. N. Postnikov, “Concerning the theory of stability of regulating systems,” Prikl. Mat. Mekh., vol. 8, pp. 246–248, 1944 (Russian).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. M. Lyapunov, “The general problem of the stability of motion,” Internat. J. Control, vol. 55, no. 3, pp. 521–790, 1992, translated into English by A. T. Fuller.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. Mansour and B. D. O. Anderson, “On the robust stability of time-varying linear systems,” in Stability Theory (Ascona, 1995), R. Jeltsch and M. Mansour, Eds., vol. 121 of Internat. Ser. Numer. Math., pp. 135–149, Birkhäuser, Basel, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Q. Mayne and H. Michalska, “Receding horizon control of nonlinear systems,” IEEE Trans. Automat. Control, vol. 35, no. 7, pp. 814–824, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. P. Molchanov and Ye. S. Pyatnitskiy, “Criteria of asymptotic stability of differential and difference inclusions encountered in control theory,” Systems Control Lett., vol. 13, no. 1, pp. 59–64, 1989.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
B. S. Mordukhovich, “Discrete approximations and refined Euler-Lagrange conditions for nonconvex differential inclusions,” SIAM J. Control Optim., vol. 33, no. 3, pp. 882–915, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Orlov, Y. Lou, and P. D. Christofides, “Robust stabilization of infinite-dimensional systems using sliding-mode output feedback control,” Internat. J. Control, vol. 77, no. 12, pp. 1115–1136, 2004.
View at: Google Scholar | MathSciNet
P. C. Parks, “A. M. Lyapunov's stability theory—100 years on,” IMA J. Math. Control Inform., vol. 9, no. 4, pp. 275–303, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
I. R. Petersen and C. V. Hollot, “A Riccati equation approach to the stabilization of uncertain linear systems,” Automatica J. IFAC, vol. 22, no. 4, pp. 397–411, 1986.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Polak, Optimization, vol. 124 of Applied Mathematical Sciences, Springer, New York, 1997.
View at: Zentralblatt MATH | MathSciNet
V. M. Popov, “Absolute stability of nonlinear systems of automatic control,” Automat. i Telemekh., vol. 22, pp. 961–979, 1961 (Russian), translated in Automat Remote Control \textbf{22} (1962), 857–875.
View at: Google Scholar
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, 1992.
View at: Zentralblatt MATH | MathSciNet
E. S. Pyatnitskii and L. B. Rapoport, “The boundary of the domain of asymptotic stability of selector-linear differential inclusions, and the existence of periodic solutions,” Soviet Math. Dokl., vol. 44, pp. 785–790, 1992.
View at: Google Scholar
B. Song and J. K. Hedrick, “Simultaneous quadratic stabilization for a class of non-linear systems with input saturation using dynamic surface control,” Internat. J. Control, vol. 77, no. 1, pp. 19–26, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
V. Veliov, “Second-order discrete approximation to linear differential inclusions,” SIAM J. Numer. Anal., vol. 29, no. 2, pp. 439–451, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Walk, Theory of Duality in Mathematical Programming, Springer, Vienna; Akademie, Berlin, 1989.
View at: Zentralblatt MATH | MathSciNet
D. Wexler, “On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation,” SIAM J. Math. Anal., vol. 11, no. 6, pp. 969–983, 1980.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. A. Yakubovich, “Solution of certain matrix inequalities occurring in the theory of automatic controls,” Dokl. Akad. Nauk USSR, vol. 143, pp. 1304–1307, 1962 (Russian), translated in Soviet Math. Dokl. \textbf{4} (1963), 620–623.
View at: Google Scholar
V. I. Zubov, Mathematical Methods for the Study of Automatic Control Systems, Pergamon Press, New York; Jerusalem Academic Press, Jerusalem, 1962.
View at: Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Vadim Azhmyakov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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