The Chetaev Theorem for Ordinary Difference Equations
DOI:
https://doi.org/10.4067/S0716-09172012000400007Keywords:
Lyapunov’s stability, Chetaev’s Theorem, equilibrium solution, instability, estabilidad de Lyapunov, teorema de Chetaev, solución de equilibrio, inestabilidad.Abstract
In this work we obtain necessary conditions for the instability in the Lyapunov sense of equilibrium points of autonomous and nonautonomous difference equations through the adaptation of differential methods and techniques due to Chetaev [3].References
[1] Agarwal, R. P., Difference Equations and Inequalities, Marcel Dekker, New York, (1992).
[2] Chetaev, N. G. Concerning the stability and instability of irregular systems. (Russian) Akad. Nauk SSSR. Prikl. Mat. Meh., 12, pp. 639642 (1948).
[3] Chetaev, N. G., The stability of motion. New York: Pergamom Press (1961).
[4] Diamond, P., Finite stability domains for difference equations, Jour. Austral. Soc., 22A, pp. 177-181 (1976).
[5] Diamond, P., Discrete Liapunov function with δ2v > 0, Jour. Austral. Soc., 20B, pp. 280-284 (1978).
[6] Elaydi, S., Asymptotics for linear difference equations I: Basic theory, J. Difference Equ. Appl., 5, pp. 563-589, (1999).
[7] Elaydi, S., An introduction to difference equations, Third Edition, Springer (2005).
[8] Lasalle, J. P., The stability and control of discrete processes, Appl. Math. Sci., 62, (1986).
[9] Lyapunov, A., Probleme general de la stabilite du movement, Ann. of Math., Study, 17, (1947).
[10] Krasovskii, N., Stability of motion, Stanford Univ. Press-Stanford, California, (1963).
[11] Perron, O., Uber einen Satz des Herrn Poincare, J. Reine Angew. Math., 136, pp. 17 37, (1909).
[12] Pituk, M., criterion for the exponential stability of linear difference equations, Appl. Math. Lett., 181, pp. 779-783, (2004).
[13] Sasu, B., Stability of difference equations and applications to robustness problems, Advances in Difference Equatiuons, ID 869608, 24 pages, (2010).
[14] Sugiyama, S. Difference inequalities and their applications to stability problem, Lectures Notes in Math., Springer, 243, pp. 1-15, (1971).
[2] Chetaev, N. G. Concerning the stability and instability of irregular systems. (Russian) Akad. Nauk SSSR. Prikl. Mat. Meh., 12, pp. 639642 (1948).
[3] Chetaev, N. G., The stability of motion. New York: Pergamom Press (1961).
[4] Diamond, P., Finite stability domains for difference equations, Jour. Austral. Soc., 22A, pp. 177-181 (1976).
[5] Diamond, P., Discrete Liapunov function with δ2v > 0, Jour. Austral. Soc., 20B, pp. 280-284 (1978).
[6] Elaydi, S., Asymptotics for linear difference equations I: Basic theory, J. Difference Equ. Appl., 5, pp. 563-589, (1999).
[7] Elaydi, S., An introduction to difference equations, Third Edition, Springer (2005).
[8] Lasalle, J. P., The stability and control of discrete processes, Appl. Math. Sci., 62, (1986).
[9] Lyapunov, A., Probleme general de la stabilite du movement, Ann. of Math., Study, 17, (1947).
[10] Krasovskii, N., Stability of motion, Stanford Univ. Press-Stanford, California, (1963).
[11] Perron, O., Uber einen Satz des Herrn Poincare, J. Reine Angew. Math., 136, pp. 17 37, (1909).
[12] Pituk, M., criterion for the exponential stability of linear difference equations, Appl. Math. Lett., 181, pp. 779-783, (2004).
[13] Sasu, B., Stability of difference equations and applications to robustness problems, Advances in Difference Equatiuons, ID 869608, 24 pages, (2010).
[14] Sugiyama, S. Difference inequalities and their applications to stability problem, Lectures Notes in Math., Springer, 243, pp. 1-15, (1971).
Published
2013-02-19
How to Cite
[1]
C. Cárcamo and C. Vidal, “The Chetaev Theorem for Ordinary Difference Equations”, Proyecciones (Antofagasta, On line), vol. 31, no. 4, pp. 391-402, Feb. 2013.
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