Find out how to access previewonly content
Date:
27 Mar 2013
On the construction of the Lyapunov function with signdefinite derivative with the help of auxiliary functions with signconstant derivatives
 Aleksandr O. Ignatyev
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The short survey of studies of the asymptotic stability with the help of auxiliary functions (positive definite and nonnegative) whose derivatives are nonpositive by virtue of the equations of perturbed motion and the construction of positive definite Lyapunov functions with negative definite derivatives on the basis of these auxiliary functions is given. The example of the construction of the Lyapunov function with the use of an auxiliary nonnegative function with nonpositive derivative is presented.
Presented by A. M. Samoilenko
Translated from Russian by V. V. Kukhtin
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 9, No. 4, pp. 515–545, October–November, 2012.
 D. Aeyels, “Asymptotic stability of nonautonomous systems by Liapunov’s direct method,” Systems & Control Lett., 25, No. 4, 273–280 (1995). CrossRef
 D. Aeyels and J. Peuteman, “A new asymptotic stability criterion for nonlinear timeinvariant differential equations,” IEEE Trans. Automat. Control, 43, No. 7, 968–971 (1998). CrossRef
 N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific, Singapore, 2006. CrossRef
 D. Angeli, “Inputtostate stability of PDcontrolled robotic systems,” Automatica, 35, No. 7, 1285–1290 (1999). CrossRef
 Yu. M. Aponin and E. A. Aponina, “The LaSalle invariance principle and mathematical models of evolution of microbial populations,” Komp. Issl. Model., 3, No. 2, 177–190 (2011).
 Z. S. Athanassov, “Total stability of sets for nonautonomous differential systems,” Trans. of AMS, 295, No. 2, 649–663 (1986). CrossRef
 A. Bacciotti and L. Mazzi, “An invariance principle for nonlinear switched systems,” Systems & Control Lett., 54, No. 11, 1109–1119 (2005). CrossRef
 A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, Springer, Berlin, 2005.
 R. Balan, “An extension of Barbashin–Krasovskii–LaSalle theorem to a class of nonautonomous systems,” Nonlin. Dyn. Syst. Theory, 8, 225–268 (2008).
 E. A. Barbashin and N. N. Krasovskii, “On the stability of motion on the whole,” Dokl. AN SSSR, 86, No. 6, 146–152 (1952).
 N. P. Bhatia and G. P. Szego, Stability Theory of Dynamical Systems, Springer, Berlin, 2002.
 A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, Amer. Inst. of Math. Sci., Springfield, 2007.
 N. F. Britton, Essential Mathematical Biology, Springer, Berlin, 2005.
 N. G. Bulgakov and B. S. Kalitin, “A generalization of theorems of Lyapunov second method. I. Theory,” Proc. of NAS of BSSR, Ser. Fiz.Mat. Nav., 3, 32–36 (1978).
 T. A. Burton, “Some Liapunov theorems,” SIAM J. Control, 4, No. 3, 460–465 (1966). CrossRef
 T. A. Burton, “An extension of Liapunov’s direct method,” J. Math. Anal. and Appl., 28, No. 3, 545–552 (1969). CrossRef
 T. A. Burton, “Correction to “an extension of Liapunov’s direct method”,” J. Math. Anal. and Appl., 32, No. 3, 689–691 (1970). CrossRef
 D. Cheng, J. Wang, and X. Hu, “An extensions of LaSalle’s invariance principle and its application to multiagent consensus,” IEEE Trans. Automat. Control, 53, No. 7, 1765–1770 (2008). CrossRef
 J. L. Corne and N. Rouche, “Attractivity of closed sets proved by using a family of Liapunov functions,” J. of Diff. Equa., 13, No. 2, 231–246 (1973). CrossRef
 A. D’Anna, “Proving conditional attractivity of a closed set with a family of Liapunov functions,” Int. J. of NonLin. Mechanics, 12, No. 3, 103–111 (1977). CrossRef
 S. M. Dobrovol’skii, A. S. Kotyurgina, and R. K. Romanovskii, “On the stability of solutions of linear systems with almost periodic matrix,” Matem. Zam., 36, No. 6, 473–489 (2002).
 L. Faubourg and J.B. Pomet, “Control Lyapunov functions for homogeneous “JurdjevicQuinn” systems,” ESAIM: Control, Optim. Calc. Variat., 5, 293–311 (2000). CrossRef
 R. I. Gladilina and A. O. Ignatyev, “On the stability of periodic impulsive systems,” Math. Notes, 76, No. 1, 41–47 (2004). CrossRef
 R. Goebel, R. G. Sanfelice, and A. R. Teel, “Invariance principles for switching systems via hybrid systems techniques,” Systems & Control Lett., 57, No. 12, 980–986 (2008). CrossRef
 E. I. Grudo, “To the etheory of stability of ordinary differential systems and Pfaff’s systems,” Diff. Uravn., 19, No. 5, 782–789 (1983).
 F. W. M. Haddad, V. Chellaboina, and S. G. Nersesov, “Hybrid nonnegative and compartmental dynamical systems,” Math. Probl. Engin., 8, No. 6, 493–515 (2002). CrossRef
 W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Univ. Press, Princeton, 2006.
 J. R. Haddock, “On Liapunov functions for nonautonomous systems,” J. of Math. Anal. Appl., 47, No. 3, 599–603 (1974). CrossRef
 J. R. Haddock and J. Terjéki, “Liapunov–Razumikhin functions and an invariance principle for functional differential equations,” J. Differ. Equa., 48, No. 1, 95–122 (1983). CrossRef
 L. Hatvani, “On partial asymptotic stability and instability. I. Autonomous systems,” Acta Sci. Math., 45, 219–231 (1983).
 J. P. Hespanha, “Uniform stability of switched linear systems: extensions of LaSalle’s invariance principle,” IEEE Trans. Automat. Control, 49, No. 4, 470–482 (2004). CrossRef
 J. Hurt, “Some stability theorems for ordinary difference equations,” SIAM J. on Numer. Anal., 4, No. 4, 582–596 (1967). CrossRef
 A. Iggidr, B. Kalitine, and R. Outbib, “Semidefinite Lyapunov functions. Stability and stabilization,” Math. of Control, Signals, and Systems, 9, No. 2, 95–106 (1996). CrossRef
 A. Iggidr and G. Sallet, “On the stability of nonautonomous systems,” Automatica, 39, No. 1, 167–171 (2003). CrossRef
 A. O. Ignatyev, “Application of Lyapunov’s direct method to the study of integral sets,” Ukr. Mat. Zh., 44, No. 10, 1342–1348 (1992).
 A. O. Ignatyev, “On the existence of Lyapunov functions in problems of the stability of integral sets,” Ukr. Mat. Zh., 45, No. 7, 932–941 (1993).
 A. O. Ignatyev, “On the stability of equilibrium for almost periodic systems,” Nonlin. Anal., 29, No. 8, 957–962 (1997). CrossRef
 A. O. Ignatyev, “On the asymptotic stability in functional differential equations,” Proc. Amer. Math. Soc., 127, No. 6, 1753–1760 (1999). CrossRef
 A. O. Ignatyev, “Study of the stability with the help of signconstant Lyapunov functions,” Ukr. Mat. Vest., 2, No. 1, 74–83 (2005).
 O. A. Ignatyev, “On the partial asymptotic stability in nonautonomous differential equations,” Diff. Integr. Equa., 19, No. 7, 831–839 (2006).
 A. O. Ignatyev and O. A. Ignatyev, “On the stability in periodic and almost periodic difference systems,” J. Math. Anal. and Appl., 313, No. 2, 678–688 (2006). CrossRef
 O. A. Ignatyev and V. Mandrekar, “BarbashinKrasovskii theorem for stochastic differential equations,” Proc. AMS, 138, No. 11, 4123–4128 (2010). CrossRef
 L. Jiang, “Asymptotic stability and instability criteria for nonautonomous systems,” J. of Math. Sci., 177, No. 3, 395–401 (2011). CrossRef
 B. S. Kalitin, “On the method of semidefinite Lyapunov functions for nonautonomous differential systems,” Diff. Uravn., 34, No. 4, 583–590 (1995).
 B. S. Kalitin, “The stability of closed invariant sets of semidynamical systems,” Diff. Uravn., 38, No. 11, 1565–1566 (2002).
 B. S. Kalitin, Mathematical FirstOrder Models of a Competitive Market [in Russian], BelGU, Minsk, 2011.
 R. E. Kalman and J. E. Bertram, “Control system analysis and design via the “second method” of Lyapunov. I. Continuoustime systems,” Trans. ASME Ser. D, 82, 371–393 (1960). CrossRef
 A. A. Kosov, “To the method of Lyapunov vector functions,” in Lyapunov Functions and Their Applications [in Russian], SO AN SSSR, Novosibirsk, 1986, pp. 106–110.
 N. N. Krasovskii, Some Problems of the Theory of Stability of Motion, Stanford Univ. Press, Stanford, 1963.
 R. Kristiansen and P. J. Nicklasson, “Spacecraft formation flying: A review and new results on state feedback control,” Acta Astronaut., 65, No. 11–12, 1537–1552 (2009). CrossRef
 M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995.
 J. P. LaSalle, “Some extensions of Liapunov’s second method,” IRE Trans. Circuit Theory, 7, No. 4, 520–527 (1960).
 J. P. LaSalle, “Stability theory for ordinary differential equations,” J. Diff. Equa., 4, No. 1, 57–65 (1968). CrossRef
 W. Leighton, “On the construction of Liapunov functions for certain autonomous nonlinear differential equations,” Contr. Diff. Equa., 2, 367–383 (1963).
 J. J. Levin and J. A. Nohel, “Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics,” Arch. Rat. Mech. Anal., 5, No. 1, 194–211 (1960). CrossRef
 B. M. Levitan, AlmostPeriodic Functions [in Russian], Gostekhteorizdat, Moscow, 1953.
 X. Liu, “On (h _{0} , h)stability of autonomous systems,” J. of Appl. Math. and Stoch. Anal., 5, No. 4, 331–338 (1992). CrossRef
 A. Loria, E. Panteley, D. Popovic, and A. Teel, “A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems,” IEEE Trans. Automat. Control, 50, No. 2, 183–198 (2005). CrossRef
 A. M. Lyapunov, Collection of Works [in Russian], Izd. AN SSSR, Moscow–Leningrad, 1956, Vol. 2.
 I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow, 1966.
 M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer, London, 2009. CrossRef
 I. L. Massera, “On Liapunoff conditions of stability,” Ann. of Math., 50, No. 3, 705–721 (1949). CrossRef
 V. M. Matrosov, “On the stability of motion,” Prikl. Mat. Mekh., 26, No. 5, 885–895 (1962).
 F. Mazenc and M. Malisoff, “Strict Lyapunov function constructions under LaSalle conditions with an application to LotkaVolterra systems,” IEEE Trans. Automat. Control, 55, No. 4, 841–854 (2010). CrossRef
 F. Mazenc, M. Malisoff, and O. Bernard, “Lyapunov functions and robustness analysis under Matrosov conditions with an application to biological systems,” in Proceed. of the American Control Conference, 2008, 2933–2938.
 F. Mazenc, M. Malisoff, and O. Bernard, “A simplified design for strict Lyapunov functions under Matrosov conditions,” IEEE Trans. Automat. Control, 54, No. 1, 177–183 (2009). CrossRef
 F. Mazenc and D. Nešić, “Strong Lyapunov functions for systems satisfying the conditions of LaSalle,” IEEE Trans. Automat. Control, 49, No. 6, 1026–1030 (2004). CrossRef
 F. Mazenc and D. Nešić, “Lyapunov functions for timevarying systems satisfying generalized conditions of Matrosov theorem,” Math. Control Sign. Syst., 19, No. 2, 151–182 (2007). CrossRef
 A. P. Mishin and I. V. Proskuryakov, Higher Agebra [in Russian], Nauka, Moscow, 1965.
 A. P. Morgan and K. S. Narendra, “On the uniform asymptotic stability of certain linear nonautonomous differential equations,” SIAM J. Control and Opt., 15, No. 1, 5–24 (1977). CrossRef
 M. A. Nowak and R. M. May, Virus Dynamics. Mathematical Principles of Immunology and Virology, Oxford Univ. Press, Oxford, 2000.
 M. N. Oguztreli, V. Lakshmikantham, and S. Leela, “An algorithm for the construction of Liapunov functions,” Nonlin. Anal., TMA, 11, No. 5, 1195–1212 (1981). CrossRef
 B. Paden and R. Panja, “Globally asymptotically stable ’PD+’ controller for robot manipulators,” Int. J. of Control, 47, No. 6, 1697–1712 (1988). CrossRef
 P. A. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD thesis, Caltech, Pasadena, CA, 2000.
 M. Reed and B. Simon, Methods of Mathematical Physics 1: Functional Analysis, Academic Press, New York, 1972.
 N. Rouche, “Attractivity of certain sets proved by using several Liapunov functions,” in Symposia Matematica, Vol. 6, New York, 1971, pp. 331–343.
 N. Rouche, “On the stability of motion,” Int. J. Nonlin. Mech., 3, 331–343 (1968). CrossRef
 R. Rouche and J. Mawhin, Ordinary Differential Equations II: Stability and Periodical Solutions, Pitman, London, 1980.
 N. Rouche, P. Habets, and M. Laloy, Lyapunov’s Direct Method in Stability Theory, Springer, New York, 1977. CrossRef
 A. M. Samoilenko, “Study of dynamical systems with the help of signconstant functions,” Ukr. Mat. Zh., 24, No. 3, 373–383 (1972).
 A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations [in Russian], Nauka, Moscow, 1987.
 R. G. Sanfelice, R. Goebel, and A. R. Teel, Results on convergence in hybrid systems via detectability and an invariance principle, American Control Conference, June 8–10, 2005, pp. 551–556.
 V. A. Sarychev, “Asymptotically stable stationary rotational motions of a satellite,” In Proc. of the 1st IFAC Symposium on automatic control in space, Stavanger, Norway, 1965, pp. 277–286.
 R. Sepulchre, M. Jankovic, and P. V. Kokotovic, Constructive Nonlinear Control, Springer, Berlin, 1997. CrossRef
 N. Sreedhar, “Concerning Liapunov functions for linear systems — I,” Int. J. of Control, 11, No. 1, 165–171 (1970). CrossRef
 J. Wang, D. Cheng, and X. Hu, “An extension of LaSalle’s invariance principle for a class of switched linear systems,” Systems & Control Lett., 58, No. 10–11, 754–758 (2009). CrossRef
 Z. M. Wang, Y. Tan, G. X. Wang, and D. Ne˘si´c, “On stability properties of nonlinear timevarying systems by semidefinite timevarying Lyapunov candidates,” in Proceed. of the 17th World Congress, July 6–11, 2008, the Federation of Automatic Control, Seoul, 2008, pp. 1123–1128.
 T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer, New York, 1975. CrossRef
 T. Yoshizawa, “Asymptotic behavior of solutions of a system of differential equations,” Contr. Differ. Equa., 1, 371–387 (1963).
 Title
 On the construction of the Lyapunov function with signdefinite derivative with the help of auxiliary functions with signconstant derivatives
 Journal

Journal of Mathematical Sciences
Volume 190, Issue 4 , pp 567588
 Cover Date
 20130401
 DOI
 10.1007/s1095801312713
 Print ISSN
 10723374
 Online ISSN
 15738795
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Asymptotic stability
 auxiliary functions with signconstant derivatives
 Lyapunov function
 theorems of Barbashin–Krasovskii and Matrosov
 construction of the Lyapunov function on their basis
 Industry Sectors
 Authors

 Aleksandr O. Ignatyev ^{(1)}
 Author Affiliations

 1. Institute of Applied Mathematics and Mechanics, NASU, 74, R. Luxemburg Str., Donetsk, 83114, Ukraine