On the Stability and Stabilization Problems of Volterra Integro-Differential Equations

    2018, Vol. 14, no. 3, pp.  387-407

    Author(s): Andreev A. S., Peregudova O. A.

    In this paper, the stability and stabilization problems for nonlinear Volterra integrodifferential equations with unlimited delay are considered. The development of the direct Lyapunov method in the study of the limiting properties of the solutions of these equations is carried out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics of these equations has been constructed revealing the limiting properties of their solutions. The assumption of the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete solution to the positive limit set localization problem. On this basis new theorems on sufficient conditions for the asymptotic stability and instability of the zero solution of nonlinear Volterra integro-differential equations are proved. These theorems are applied to the problem of the equilibrium position stability of the hereditary mechanical systems as well as the regulation problem of the controlled mechanical systems using a proportional-integro-differential controller. As an example, the regulation problem of a mobile robot with three omnidirectional wheels and a displaced mass center is solved using the nonlinear integral controllers without velocity measurements.
    Keywords: Volterra integro-differential equation, stability, Lyapunov functional, limiting equation, regulation problem
    Citation: Andreev A. S., Peregudova O. A., On the Stability and Stabilization Problems of Volterra Integro-Differential Equations, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp.  387-407

    Download File
    PDF, 369.81 Kb


    [1] Alexandrov, A. G. and Palenov, M. V., “Adaptive PID Controllers: State of the Art and Development Prospects”, Autom. Remote Control, 75:2 (2014), 188–199  mathnet  crossref  mathscinet; Avtomat. i Telemekh., 2014, no. 2, 16–30 (Russian)
    [2] Anan'evsky, I. M. and Kolmanovskii, V. B., “On Stabilization of Some Control Systems with an After-Effect”, Autom. Remote Control, 50:9, part 1 (1989), 1174–-1181  mathscinet; Avtomat. i Telemekh., 1989, no. 9, 34–43 (Russian)  mathnet
    [3] Andreev, A. S., “On the Asymptotic Stability and Instability of the Zeroth Solution of a Non-Autonomous System”, J. Appl. Math. Mech., 48:2 (1984), 154–160  crossref  mathscinet; Prikl. Mat. Mekh., 48:2 (1984), 225–232 (Russian)  mathscinet
    [4] Andreev, A. S., “The Lyapunov Functionals Method in Stability Problems for Functional Differential Equations”, Autom. Remote Control, 70 (2009), 1438–-1486  mathnet  crossref  mathscinet  zmath; Avtomat. i Telemekh., 2009, no. 9, 4–-55 (Russian)  zmath
    [5] Andreev, A. S. and Peregudova, O. A., “On the Method of Comparison in Asymptotic-Stability Problems”, Dokl. Phys., 50:2 (2005), 91–94  mathnet  crossref  mathscinet  adsnasa; Dokl. Akad. Nauk, 400:5 (2005), 621–624 (Russian)
    [6] Andreyev, A. S. and Peregudova, O. A., “The Comparison Method in Asymptotic Stability Problems”, J. Appl. Math. Mech., 70:6 (2006), 865–875  crossref  mathscinet; Prikl. Mat. Mekh., 70:6 (2006), 965–976 (Russian)  mathscinet
    [7] Andreev, A. S. and Peregudova, O. A., “Stabilization of the Preset Motions of a Holonomic Mechanical System without Velocity Measurement”, J. Appl. Math. Mech., 81:2 (2017), 95–105  crossref  mathscinet; Prikl. Mat. Mekh., 81:2 (2017), 137–153 (Russian)
    [8] Andreev, A. S. and Peregudova, O. A., “Non-Linear PI Regulators in Control Problems for Holonomic Mechanical Systems”, Systems Sci. Control Eng., 6:1 (2018), 12–19  crossref
    [9] Andreev, A. S., Peregudova, O. A., and Rakov, S. Yu., “On Modeling a Nonlinear Integral Regulator on the Base of the Volterra Equations”, Zh. Srednevolzhsk. Mat. Obshch., 18:3 (2016), 8–18 (Russian)  mathscinet
    [10] Åström, K. J. and Hägglund, T., Advaced PID Control, ISA, Research Triangle Park, N.C., 2006, 460 pp.
    [11] Athanassov, Zh. S., “Families of Liapunov – Krasovskii Functionals and Stability for Functional Differential Equations”, Ann. Mat. Pura Appl. (4), 176 (1999), 145–165  crossref  mathscinet  zmath
    [12] Borisov, A. V., Kilin, A. A., and Mamaev, I. S., “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172  mathnet  crossref  mathscinet  zmath  adsnasa
    [13] Burton, T. A., “Stability Theory for Delay Equations”, Funkcial. Ekvac., 22:1 (1979), 67–76  mathscinet  zmath
    [14] Burton, T. A., Volterra Integral and Differential Equations, Math. Sci. Eng., 202, 2nd ed., Elsevier, Amsterdam, 2005, x+353 pp.  mathscinet  zmath
    [15] Burton, T. A. and Zhang, S. N., “Unified Boundedness, Periodicity, and Stability in Ordinary and Functional-Differential Equations”, Ann. Mat. Pura Appl. (4), 145 (1986), 129–158  crossref  mathscinet  zmath
    [16] Bykov, Ya. V., On Some Problems of the Theory of Integro-Differential Equations, KGU, Frunze, 1957, 327 pp. (Russian)
    [17] Coleman, B. D. and Dill, E. H., “On the Stability of Certain Motions of Incompressible Materials with Memory”, Arch. Rational Mech. Anal., 30:3 (1968), 197–224  crossref  mathscinet  zmath  adsnasa
    [18] Coleman, B. D. and Mizel, V. J., “On the Stability of Solutions of Functional Differential Equations”, Arch. Rational Mech. Anal., 30:3 (1968), 173–196  crossref  mathscinet  zmath  adsnasa
    [19] Coleman, B. D. and Owen, D. R., “On the Initial Value Problem for a Class of Functional-Differential Equations”, Arch. Rational Mech. Anal., 55:4 (1974), 275–299  crossref  mathscinet  zmath  adsnasa
    [20] Corduneanu, C. and Lakshmikantham, V., “Equations with Unbounded Delay: A Survey”, Nonlinear Anal., 4:5 (1980), 831–877  crossref  mathscinet  zmath
    [21] Haddock, J., Krisztin, T., and Terjéki, J., “Invariance Principles for Autonomous Functional-Differential Equations”, J. Integral Equations, 10:1–3, suppl. (1985), 123–136  mathscinet  zmath
    [22] Haddock, J. and Terjéki, J., “On the Location of Positive Limit Sets for Autonomous Functional Differential Equations with Infinite Delay”, J. Differential Equations, 86:1 (1990), 1–32  crossref  mathscinet  zmath  adsnasa
    [23] Filatov, A. N., Averaging Methods in Differential and Integro-Differential Equations, Fan, Tashkent, 1971, 279 pp. (Russian)  mathscinet  zmath
    [24] Hale, J. K., “Sufficient Conditions for the Stability and Instability of Autonomous Functional Differential Equations”, J. Differential Equations, 1:4 (1965), 452–482  crossref  mathscinet  zmath  adsnasa
    [25] Hale, J. K., Theory of Functional Differential Equations, Appl. Math. Sci., 3, Springer, New York, 1977, X, 366 pp.  crossref  mathscinet  zmath
    [26] Hale, J. K. and Kato, J., “Phase Space for Retarded Equations with Infinite Delay”, Funkcial. Ekvac., 21:1 (1978), 11–41  mathscinet  zmath
    [27] Hino, Y., “On Stability of the Solution of Some Functional Differential Equations”, Funkcial. Ekvac., 14 (1971), 47–60  mathscinet  zmath
    [28] Hino, Y., “Stability Properties for Functional Differential Equations with Infinite Delay”, Tôhoku Math. J., 35:4 (1983), 597–605  crossref  mathscinet  zmath
    [29] Hino, Y., Murakami, S., and Naito, T., Functional Differential Equations with Infinite Delay, Lecture Notes in Math., 1473, Springer, New York, 1991, x+317 pp.  crossref  mathscinet  zmath
    [30] Hornor, W., “Invariance Principles and Asymptotic Constancy of Solutions of Precompact Functional Differential Equations”, Tôhoku Math. J., 42:2 (1990), 217–229  crossref  mathscinet  zmath
    [31] Kato, J., “Stability Problems in Functional Differential Equations with Infinite Delay”, Funkcial. Ekvac., 21:1 (1978), 63–80  mathscinet  zmath
    [32] Kato, J., “Liapunov's Second Method in Functional Differential Equations”, Tôhoku Math. J. (2), 32:4 (1980), 487–497  crossref  mathscinet  zmath
    [33] Kato, J., “Asymptotic Behavior in Functional-Differential Equations with Infinite Delay”, Equadiff 82 (Würzburg, 1982), Lecture Notes in Math., 1017, Springer, Berlin, 1983, 300–312  crossref  mathscinet
    [34] Kerimov, M. K., “A Bibliography of Some New Papers on Integral and Integro-Differential Equations”, V. Volterra. Theory of Functionals and of Integral and Integro-Differential Equations, Nauka, Moscow, 1982, 257–302 (Russian)  mathscinet
    [35] Krasovskii, N. N., Stability of Motion: Applications of Lyapunov's Second Method to Differential Systems and Equations with Delay, Stanford Univ. Press, Stanford, 1963, 192 pp.  mathscinet  zmath
    [36] LaSalle, J., “The Extent of Asymptotic Stability”, Proc. Natl. Acad. Sci. USA, 46:3 (1960), 363–365  crossref  mathscinet  zmath  adsnasa
    [37] Makay, G., “On the Asymptotic Stability of the Solutions of Functional-Differential Equations with Infinite Delay”, J. Differential Equations, 108:1 (1994), 139–151  crossref  mathscinet  zmath  adsnasa
    [38] Martynenko, Yu. G., “Stability of Steady Motions of a Mobile Robot with Roller-Carrying Wheels and a Displaced Centre of Mass”, J. Appl. Math. Mech., 74:4 (2010), 436–442  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 74:4 (2010), 610–619 (Russian)  zmath
    [39] Meza, J. L., Santibáñez, V., Soto, R., Perez, J., and Perez, J., “Analysis via Passivity Theory of a Class of Nonlinear PID Global Regulators for Robot Manipulators”, Advances in PID Control, ed. V. D. Yurkevich, InTech, Rijeka, 2011, 45–-64
    [40] Miller, R., “Asymptotic Behavior of Solutions of Nonlinear Differential Equations”, Trans. Amer. Math. Soc., 115 (1965), 400–416  crossref  mathscinet  zmath
    [41] Murakami, S., “Perturbation Theorems for Functional Differential Equations with Infinite Delay via Limiting Equations”, J. Differential Equations, 59:3 (1985), 314–335  crossref  mathscinet  zmath  adsnasa
    [42] Murakami, S. and Naito, T., “Fading Memory Spaces and Stability Properties for Functional Differential Equations with Infinite Delay”, Funkcial. Ekvac., 32:1 (1989), 91–105  mathscinet  zmath
    [43] O’Dwyer, A., Handbook of PI and PID Controller Tuning Rules, 3rd ed., Imperial College Press, London, 2009, 624 pp.
    [44] Pavlikov, S. V., “On Stabilization of the Controlled Mechanical Systems”, Autom. Remote Control, 68:9 (2007), 1482–-1491  mathnet  crossref  mathscinet  zmath; Avtomat. i Telemekh., 2007, no. 9, 16–-26 (Russian)  mathscinet  zmath
    [45] Pavlikov, S. V., “On the Stability of the Motions of Hereditary Systems with Infinite Delay”, Dokl. Math., 76:2 (2007), 678–680  crossref  mathscinet  zmath; Dokl. Akad. Nauk, 416:2 (2007), 166–168 (Russian)  mathnet  mathscinet  zmath
    [46] Pavlikov, S. V., “On the Problem of the Stability of Functional-Differential Equations with Infinite Delay”, Russian Math. (Iz. VUZ), 52:7 (2008), 24–32  mathnet  crossref  mathscinet  zmath; Izv. Vyssh. Uchebn. Zaved. Mat., 2008, no. 7, 29–38 (Russian)  mathscinet  zmath
    [47] Peregudova, O. A., “Development of the Lyapunov Function Method in the Stability Problem for Functional-Differential Equations”, Differ. Equ., 44:12 (2008), 1701–-1710  crossref  mathscinet  zmath; Differ. Uravn., 44:12 (2008), 1638–-1647 (Russian)  mathscinet
    [48] Rouche, N., Habets, P., and Laloy, M., Stability Theory by Lyapunov’s Direct Method, Appl. Math. Sci., 22, Springer, New York, 1977, XII, 396 pp.  crossref  mathscinet
    [49] Sawano, K., “Exponential Asymptotic Stability for Functional-Differential Equations with Infinite Retardations”, Tôhoku Math. J. (2), 31:3 (1979), 363–382  crossref  mathscinet  zmath
    [50] Sawano, K., “Positively Invariant Sets for Functional Differential Equations with Infinite Delay”, Tôhoku Math. J. (2), 32:2 (1980), 557–566  crossref  mathscinet  zmath
    [51] Sawano, K., “Some Considerations on the Fundamental Theorems for Functional-Differential Equations with Infinite Delay”, Funkcial. Ekvac., 25:1 (1982), 97–104  mathscinet  zmath
    [52] Schumacher, K., “Existence and Continuous Dependence for Functional-Differential Equations with Unbounded Delay”, Arch. Rational Mech. Anal., 67:4 (1978), 315–335  crossref  mathscinet  zmath  adsnasa
    [53] Sell, G. R., Topological Dynamics and Ordinary Differential Equations, v. 33, Van Nostrand Reinhold Mathematical Studies, Van Nostrand Reinhold, London, 1971  mathscinet  zmath
    [54] Sergeev, V. S., “The Stability of the Equilibrium of a Wing in an Unsteady Flow”, J. Appl. Math. Mech., 64:2 (2000), 219–228  crossref  mathscinet  zmath; Prikl. Mat. Mekh., 64:2 (2000), 219–228 (Russian)  mathscinet  zmath
    [55] Sergeev, V. S., “Stability of Solutions of Volterra Integrodifferential Equations”, Math. Comput. Modelling, 45:11–12 (2007), 1376–-1394  crossref  mathscinet  zmath
    [56] Sergeev, V. S., “Resonance Oscillations in Some Systems with Aftereffect”, J. Appl. Math. Mech., 79:5 (2015), 432–439  crossref  mathscinet; Prikl. Mat. Mekh., 79:5 (2015), 615–626 (Russian)
    [57] Volterra, V., Theory of Functionals and of Integral and Integro-Differential Equations, Dover, New York, 1959  mathscinet  zmath
    [58] Wakeman, D. R., “An Application of Topological Dynamics to Obtain a New Invariance Property for Nonautonomous Ordinary Differential Equations”, J. Differential Equations, 17 (1975), 259–295  crossref  mathscinet  zmath  adsnasa

    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License