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References

  • Anashkin, O.V. (1978), On investigations of stability under persistently acting disturbances in the neutral case. Different Uravn, 14, No.6, 1124–1127 (in Russian).

    Google Scholar 

  • Aoki, M. (1968), Control of large-scale dynamic systems by aggregation. IEEE Trans. on Aut. Control, AC-13, No.3, 246–253.

    Google Scholar 

  • Araki, M. (1975), Application of M-matrices to the stability problems of composite dynamical systems. J. Math. Analysis and Applications, 52, 309–321.

    Google Scholar 

  • Araki, M. (1978), Stability of large-scale non linear systems-quadratic-order theory of composite-system method using M-matrices. IEEE Trans. on Aut. Control, AC-23, No.2, 129–142.

    Google Scholar 

  • Araki, M., and B. Kondo (1972), Stability and transient behaviour of composite systems. IEEE Trans. on Aut. Control, AC-17, No.4, 537–541.

    Google Scholar 

  • Bailey, F.N. (1966), The application of Lyapunov's second method to interconnected systems. J. SIAM Control, Ser. A, 3, No. 3, 443–462.

    Google Scholar 

  • Bailey, F.N. (1968), The concept of aggregation in system stability analysis. 2nd Asilomar Conf. on Circ. and Syst., 570–576.

    Google Scholar 

  • Bellman, R. (1962), Vector Liapunov functions. J. SIAM Control, Ser.A, 1, No. 1, 32–34.

    Google Scholar 

  • Bitsoris, G., and C. Burgat (1976), Stability conditions and estimates of the stability region of complex systems. Int. J. Systems Science, 7, No. 8, 911–928.

    Google Scholar 

  • Blight, J.D., and N.H. McClamroch (1975), Graphical stability criteria for large-scale nonlinear multiloop systems. Preprints of the 6th IFAC World Congress, Part 44.5: 1–6.

    Google Scholar 

  • Bogoljubov, N.N., and Yu.A. Mitropolsky (1974), Asymptotic Methods in the Theory of Non-Linear Oscillations. Nauka, Moscow (in Russian).

    Google Scholar 

  • Bondi, P., P. Fergola, and L. Gambardella (1979), Partial stability of large-scale systems. IEEE Trans. on Aut. Control, AC-24, No. 1, 94–96.

    Google Scholar 

  • Borne, P., and M. Benrejeb (1977a), Stability of non-linear composite systems. 1st World Conf. on Math. at the Service of Man, Barcelona, Tech. Session 4/6, 1–12.

    Google Scholar 

  • Borne, P., and M. Benrejeb (1977b), On the stability of a class of interconnected systems. Application to the forced working condidions. 4th IFAC Symp. on Multivariable Technological Systems, Frederictown, 261–265.

    Google Scholar 

  • Borne, P., M. Benrejeb, and J.L. Cocquerelle (1978), On the synthesis of a class of interconnected systems with structural variations — application to an electrical power plant. MECO 78, IASTED, Acta Press, 1–4.

    Google Scholar 

  • Burgat, C., J. Bernussou, Lj.T. Grujić, J.C. Gentina, and P. Borne (1978), Sur la stabilité des systèmes de grande dimension. Les perturbations structurelles arbitraires et périodiques. R.A.I.R.O. Automatique/Systems Analysis and Control, 12, No. 3, 245–267.

    Google Scholar 

  • Burgat, C., and Lj.T. Grujić (1979), Sur la stabilité asymptotique d'un système de grande dimension admettant un système agrégé du type Lotka-Volterra. C.R. Acad. Sc. Paris, 288, Ser.A, 691–693.

    Google Scholar 

  • Chaplygin, S.A. (1919), New method of integration of a general differential equation of the train motion. Byul. Nauch.-Eksperim. in-ta Putej Soobshcheniya, vyp. 1a, No.9, 308–334 (in Russian).

    Google Scholar 

  • Evans, F.J. (1978), Prospect for dynamic security monitoring in large-scale electric power systems. Proc. 7th World IFAC Congress, Helsinki, 1–14.

    Google Scholar 

  • Fiedler, M., and V. Ptak (1962), On matrices with non-positive off-diagonal elements and positive principal minors. Czech. Nat. J., 12, No.87, 382–400.

    Google Scholar 

  • Gentina, J.C., P. Borne, C. Burgat, J. Bernussou, and Lj.T. Grujić (1979), Sur la stability des systèmes de grande dimension. Normes vectorielles. R.A.I.R.O. Automatique/Systems Analysis and Control, 13, No.1, 57–75.

    Google Scholar 

  • Gentina, J.C., P. Borne, and F. Laurent (1972), Stabilité des systèmes continus non linéaires de grande dimension. R.A.I.R.O., 6, No.3,69–77.

    Google Scholar 

  • Gentina, J.C., P. Borne, and F. Laurent (1973), On a study of stability and sensitivity of large continuous non-linear systems under uncertainty. 3rd IFAC Symp. on Sens. Adapt. and Optim., ISA, 412–417.

    Google Scholar 

  • Goina, L.N., and A.A. Martynyuk (1974), Systems of oscillators analysis with weak interconnection in the neighbourhood of a special point. Matem. Fizika, 15, — (in Russian).

    Google Scholar 

  • Grujić, Lj.T. (1972), Large-scale systems stability. Dissertation (published in 1974), Faculty of Mechanical Engineering, Belgrade (in Serbo-Coratian).

    Google Scholar 

  • Grujić, Lj.T. (1974a), Stability analysis of large-scale systems with stable and unstable subsystems. Int. J. Control, 20, No.3, 453–463.

    Google Scholar 

  • Grujić, Lj.T. (1974b), On multi-level absolute stability analysis of large-scale systems: the Popov method, the comparison principle and time-invariant systems. Automatika, Nos. 1–2, 67–72.

    Google Scholar 

  • Grujić, Lj.T. (1974c), On multi-level absolute stability analysis of large-scale systems: the Lyapunov method, the comparison principle and time varying systems. Automatika, Zagreb, Nos.4–5, 155–161.

    Google Scholar 

  • Grujić, Lj.T. (1975), Stability of product sets. Proc. 1975 Midwest Symposium on Circuits and Systems, Concordia University, Montreal, 254–258.

    Google Scholar 

  • Grujić, Lj.T. (1976a), Time-varying sets, aggregation and stability of large-scale systems. IEEE Proc. on Int. Symp. Circ. and Systems, Münich, 392–401.

    Google Scholar 

  • Grujić, Lj.T. (1976b), General stability analysis of large-scale systems. IFAC Symp. on Large-Scale Systems Theory and Applications, Udine, 203–213.

    Google Scholar 

  • Grujić, Lj.T. (1977a), Stability and instability of product sets. Systems Science, 3, No.1, 13–31.

    Google Scholar 

  • Grujić, Lj.T. (1977b), Un lemme matriciel réciproque; application à la stabilité absolue. C.R. Acad. Sci. Paris, Ser.A, 284, 1409–1412.

    Google Scholar 

  • Grujić, Lj.T., and C. Burgat (1979), Estimations Ei du domaine de stabilité pour un système interconnecté de comparison du type Lotka-Volterra. C.R. Acad. Sc. Paris, Ser.A, 288, 745–747.

    Google Scholar 

  • Grujić, Lj.T., and C. Burgat (1980a), Lotka-Volterra-like approach to large-scale systems stability. Int. J. Systems Sci., 11, No.10, 1131–1144.

    Google Scholar 

  • Grujić, Lj.T., and C. Burgat (1980b), Stability analysis of large-scale generalized Lotka-Volterra systems. Ricerche di Automatica, 9, No.2, 161–170.

    Google Scholar 

  • Grujić, Lj.T., J.C. Gentina, and P. Borne (1976), General aggregation of large-scale systems by vector Lyapunov functions and vector norms. Int. J. Control, 24, No.4, 529–550.

    Google Scholar 

  • Grujić, Lj.T., J.C. Gentina, P. Borne, C. Burgat, and J. Bernussou (1978), Sur la stabilité des systèmes de grande dimension. Fonctions de Lyapunov vectorielles. R.A.I.R.O. Automatique/Systems Analysis and Control, 12, No.4, 319–348.

    Google Scholar 

  • Grujić, Lj.T., and M. Ribbens-Pavella (1978), Relaxed large-scale systems stability analysis applied to power systems. IFAC VII World Congress, 27–34.

    Google Scholar 

  • Grujić, Lj.T., and M. Ribbens-Pavella (1979), Asymptotic stability of large-scale systems with application to power systems. I: domain estimation. Electrical Powers & Energy Systems, 1, No. 3, 151–157.

    Google Scholar 

  • Grujić, Lj.T., and D.D. Šiljak (1972), Stability of large-scale systems with stable and unstable subsystems. Proc. 1972 JACC, Stanford University, Paper 17-3, 550–555.

    Google Scholar 

  • Grujié, Lj.T., and D.D. Šiljak (1973), Asymptotic stability and instability of large-scale systems. IEEE Trans. on Aut. Control, AC-18, No.6, 636–645.

    Google Scholar 

  • Hahn, W. (1967), Stability of Motion. Springer Verlag, Berlin.

    Google Scholar 

  • Hatvani, L. (1975), On application of differential inequalities to the theory of stability. Vestn. Mosk. un-ta. Ser. Matem. i Mekhanika, No. 3, 83–89 (in Russian).

    Google Scholar 

  • Hatvani, L. (1980), On the continuation of solutions of differential equations by vector Lyapunov functions. Proc. Amer. Math. Soc., 79, No. 1, 59–62.

    Google Scholar 

  • Ikeda, M., and D.D. Šiljak (1980), Decentralized stabilization of linear time-varying systems. IEEE Trans. on Aut. Control, AC-25, No. 1, 106–107.

    Google Scholar 

  • Kamke, E. (1932), Zur Theorie der Systeme Gewöhnlicher Differential-Gleichungen II. Acta Mathematica, 58, 57–85.

    Google Scholar 

  • Kayande, A.A., and V. Lakshmikantham (1966), Conditionally invariant sets and vector Lyapunov functions. J. Math. Anal. and Applications, 14, 285–293.

    Google Scholar 

  • Khapayev, M.M. (1967), On a Liapunov like theorem. Dokl. AN SSSR, 176, No. 6, 1262–1265 (in Russian).

    Google Scholar 

  • Kloeden, P.E. (1975), Aggregation-decomposition and equi-ultimate boundedness. J. Australian Math. Soc., XIX, Series B, Part 2, 249–258.

    Google Scholar 

  • Kosolapov, V.I. (1979), To complex systems stability. Prykl. Mekhanika, 25, No.7, 133–136 (in Russian).

    Google Scholar 

  • Kuhtenko, A.I. (1968), Basic problems of the complex systems control theory. In Complex Control Systems, Inst. Kibernetiki AN SSSR, Kiev, 3–62 (in Russian).

    Google Scholar 

  • Ladde, G.S., and D.D. Šiljak (1975), Stochastic stability and instability of model ecosystems. 6th IFAC World Congress, Boston, 55.4, 1–7.

    Google Scholar 

  • Lakshmikantham, V. (1975), Vector Lyapunov functions and conditional stability. J. Math. Anal. and Applications, 10, 368–377.

    Google Scholar 

  • Liapunov, A.M. (1893), Investigations of a singular case of the problem of motion stability. Mat. Sbor., 17, No.2, 253–333 (in Russian).

    Google Scholar 

  • Martynyuk, A.A. (1972a), On instability of the equilibrium position of a multidimensional system composed of “neutrally stable subsystems”, Prykl. Mekhanika, 8, No. 6, 77–82.

    Google Scholar 

  • Martynyuk, A.A. (1972b), On a Liapunov like theorem on stability of a multidimensional system. Ukr. Mat. Sb., 24, No.4, 532–537 (in Russian).

    Google Scholar 

  • Martynyuk, A.A. (1973), Qualitative investigations of behaviour of weakly coupled oscillators in a vicinity of the equilibrium position. Prykl. Mekhanika, 9, No.7, 122–126 (in Russian).

    Google Scholar 

  • Martynyuk, A.A. (1975a), Stability of Motion of Complex Systems, Naukova dumka, Kiev (in Russian).

    Google Scholar 

  • Martynyuk, A.A. (1975b), Decomposition and aggregation in the systems analysis. Teoret. i Prikl Mekhanika, No.1, 87–93 (in Russian).

    Google Scholar 

  • Martynyuk, A.A. (1979), On development of the Liapunov functions in the theory of stability of complex systems. Prykl. Mekhanika, 15, No. 10, 3–23 (in Russian).

    Google Scholar 

  • Martynyuk, A.A., and R. Gutowski (1979), Integral Inequalities and Stability of Motion. Naukova Dumka, Kiev (in Russian).

    Google Scholar 

  • Martynyuk, A.A., and V.I. Kosolapov (1978), The comparison principle and the averaging method in the problem of stability of not-asymptotically stable motions under persistent action of disturbances. Preprint: AN Ukr. SSR, Inst. Matem., No.33, 1–24 (in Russian).

    Google Scholar 

  • Martynyuk, A.A., and N.V. Nikitina (1981), On nonlinear systems of comparison in the problems of stability of large-scale systems. Prikl. Mekhanika, 17, No.12, 97–102 (in Russian).

    Google Scholar 

  • Matrosov, V.M. (1962), On the theory of stability of motion. Prikl. Matem. i Mekhanika, 26, 92–100 (in Russian).

    Google Scholar 

  • Matrosov, V.M. (1963), On the theory of stability of motion, II. Trudy Kazan. Aviac. Inst., 80, 22–33 (in Russian).

    Google Scholar 

  • Matrosov, V.M. (1968a), Comparison principle and vector Liapunov functions, I. Diff. Uravn., IV, No. 8, 1374–1386 (in Russian).

    Google Scholar 

  • Matrosov, V.M. (1968b), Comparison principle and vector Liapunov functions, II. Ibid., IV, No.10, 1739–1752 (in Russian).

    Google Scholar 

  • Matrosov, V.M. (1971), Vector Lyapunov functions in the analysis of nonlinear interco-nected systems. Symp. Mathematica, Bologna, VI, 209–242.

    Google Scholar 

  • Matrosov, V.M. (1972), The method of vector Lyapunov functions in feed-back systems. Autom. i Telemekh., No.9, 63–75 (in Russian).

    Google Scholar 

  • Michel, A.N. (1973), Stability analysis of interconnected systems. Berichte der Mathematisch — Statistischen Sektion, No.4, Graz., 1–107.

    Google Scholar 

  • Michel, A.N. (1974), Stability analysis of interconnected systems. J. SIAM Control, 12, No.3, 554–579.

    Google Scholar 

  • Michel, A.N., and R.K. Miller (1977), Qualitative Analysis of Large-Scale Dynamical Systems. Academic Press, New York.

    Google Scholar 

  • Michel, A.N., and D.W. Porter (1970), Stability analysis of composite systems. IEEE Trans. Aut. Control, AC-17, No.2, 222–226.

    Google Scholar 

  • Morari, M., G. Stephanopoulos, and R. Aris (1977), Finite stability regions for large-scale systems with stable and unstable subsystems. Int. J. Control., 26, No.5, 805–815.

    Google Scholar 

  • Oziraner, A.S., and V.V. Rumyantsev (1972), Method of Liapunov functions in the problem of stability of motion concerning a part of variables. Matem. i Mekhanika, 36, No.2, 364–384 (in Russian).

    Google Scholar 

  • Peiffer, K., and N. Rouche (1969), Liapunov's second method applied to partial stability. J. de Mécanique, 8, No.2, 323–334.

    Google Scholar 

  • Piontkovskii, A.A., and L.D. Rutkovkaya (1967), Investigation of certain stability theory problems by vector Lyapunov function method. Autom. i Telemekh., No.10, 23–31 (in Russian).

    Google Scholar 

  • Porter, D.W., and A.N. Michel (1970), Stability of composite systems. Proc. of 4th Asilomar Conf. on Circuits, Systems and Computers, 634–638.

    Google Scholar 

  • Porter, D.W., and A.N. Michel (1971), Stability analysis of composite systems with nonlinear interconnections. Proc. of 1971 Midwest Symp., 6.6-1/6.6-10.

    Google Scholar 

  • Robert, F. (1964), Normes vectorielles de vecteurs et de matrices. R.F.T.I. — Chiffres, 17, No.4, 261–299.

    Google Scholar 

  • Rouche, N., P. Habets, and M. Laloy (1980), Stability Theory via Direct Liapunov's Method. MIR, Moscow (in Russian).

    Google Scholar 

  • Rouche, N., and K. Peiffer (1967), Le théorème de Lagrange-Dirichlet et la deuxième méthode de Liapounoff. Annales de la Société Scientifique de Bruxelles, 81, 19–33.

    Google Scholar 

  • Rumyantsev, V.V. (1957), To stability of motion with respect to a part of variables. Vestn. Mosk. un-ta, Ser. Matem. i Mekhanika, No.4, 9–16 (in Russian).

    Google Scholar 

  • Saeki, M., M. Araki, and B. Kondo (1980), Local stability of composite systems-frequency-domain condition and estimate of the domain of attraction. IEEE Trans. on Aut. Control, AC-25, No. 5, 936–940.

    Google Scholar 

  • Sandell, N.R., P. Varaiya, M. Athans, and M.G. Safonov (1978), Survey of decentralized control methods for large scale systems. IEEE Trans. on Aut. Control, AC-23, No. 2, 108–128.

    Google Scholar 

  • Šiljak, D.D. (1971), On large-scale system stability. Proc. 9th Ann. Allerton Conf. on Circ. and Syst. Theory, University of Illinois, 731–741.

    Google Scholar 

  • Šiljak, D.D. (1972a), Stability of large-scale systems. Proc. 1972 IFAC 5th World Congress, ISA, Pittsburgh, Paper C-32, 1–11.

    Google Scholar 

  • Šiljak, D.D. (1972b), Stability of large-scale systems under structural perturbations. IEEE Trans. SMC, SMC-2, No. 2, 657–663.

    Google Scholar 

  • Šiljak, D.D. (1973a), Competitive economic systems: stability decomposition and aggregation. IEEE Conf. Dec. and Control, 265–275.

    Google Scholar 

  • Šiljak, D.D. (1973b), On stability of large-scale systems under structural perturbations. IEEE Trans. Syst. Man and Cybern., SMC-3, 415–417.

    Google Scholar 

  • Šiljak, D.D. (1975a), On stability of the arms race. Report No. NGR; 05-017-010-7508, University of Santa Clara, NSF Conf. on Control Theory in Intern. Relations Research, Indian University, 1–44.

    Google Scholar 

  • Šiljak, D.D. (1975b), When is a complex ecosystem stable? Mathematical Biosciences, 25, 25–50.

    Google Scholar 

  • Šiljak, D.D. (1977a), On pure structure of dynamic systems. Nonlinear Analysis; Theory, Methods and Applications, 1, 397–413.

    Google Scholar 

  • Šiljak, D.D. (1977b), On the stability of the arms race. Chapter 9 in Mathematical Systems in International Relations Research, J.V. Gillespie and D.A. Zinnes, Praeger Publ., N.Y.

    Google Scholar 

  • Šiljak, D.D. (1978), Large-Scale Dynamic Systems: Stability and Structure. North-Holland, New York.

    Google Scholar 

  • Šiljak, D.D., and M.B. Vukčević (1974), On hierarchic stabilization of large-scale linear systems. 8th Asilomar Conf. on Circuits, Systems and Computers, Pacific Grove, 503–507.

    Google Scholar 

  • Sinha, A.S.C. (1980), Lyapunov functions for a class of large-scale systems. IEEE Trans. Aut. Cont., AC-25, No. 3, 558–560.

    Google Scholar 

  • Thompson, W.E. (1972), Exponential stability of interconnected systems. IEEE Trans. Aut. Control, AC-17, No. 2, 222–226.

    Google Scholar 

  • Thompson, W.E., and H.E. Koening (1972), Stability of a class of interconnected systems. Int. J. Control, 15, No. 4, 751–763.

    Google Scholar 

  • Vakhonina, G.S., A.S. Zemlyakov, and V.M. Matrosov (1973), On techniques for construction of Lyapunov vector functions for linear systems. Autom. i Telem., No. 2, 5–16 (in Russian).

    Google Scholar 

  • Vidyasagar, M. (1980), Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability. IEEE Trans. Aut. Control, AC-25, No. 4, 773–778.

    Google Scholar 

  • Weissenberger, S. (1973), Stability regions of large-scale systems. Automatica, 9, No. 5, 653–663.

    Google Scholar 

  • Wažewski, T. (1950), Systèmes des équations et des inégalités différentielles ordinaires aux deuxièmes membres monotones et leurs applications. Annales de la Société Polonaire de Mathématiques, 23, 112–166.

    Google Scholar 

  • Zadorozhny, V.F. (1969), Asymptotic stability analysis of many-dimensional systems by means of the method of vector Liapunov functions. Kibernetika i Vychislitel'naya Tekhnika, No. 1, 92–97 (in Russian).

    Google Scholar 

  • Zadorozhny, V.F., and A.A. Martynyuk (1972), Estimation of influence of connections between sub-systems on stability of a linear non-stationary system. Prikl. Mekhanika, 18, No. 9, 65–71 (in Russian).

    Google Scholar 

  • Zadorozhny, V.F., and A.A. Martynyuk (1973a), On a general problem of aggregation solution as the problem of moments. Mathematicheskaya Fizika, No. 14, 49–54 (in Russian).

    Google Scholar 

  • Zadorozhny, V.F., and A.A. Martynyuk (1973b), On solution of the general aggregation problem as the moments problem. Mat. Physics, 14, 49–55.

    Google Scholar 

  • Zadorozhny, V.F., and A.A. Martynyuk (1975), The direct Liapunov method and L-problem of moments in the problems on stability of many-dimensional systems. In Problems of Analytical Mechanics, Theories of Stability and Control, Nauka, 151–154 (in Russian).

    Google Scholar 

  • Zemlyakov, A.S. (1972), On a problem of the comparison system construction. Tr. Kazan. Aviats. in-ta, No. 144, 46–54 (in Russian).

    Google Scholar 

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(1987). Large-scale systems in general. In: Grujić, L.T., Martynyuk, A.A., Ribbens-Pavella, M. (eds) Large Scale Systems Stability under Structural and Singular Perturbations. Lecture Notes in Control and Information Sciences, vol 92. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0006853

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