The Chetaev Theorem for Ordinary Difference Equations

  • Cristian Cárcamo Universidad del Bío-Bío.
  • Claudio Vidal Universidad del Bío-Bío.
Palabras clave: Lyapunov’s stability, Chetaev’s Theorem, equilibrium solution, instability, estabilidad de Lyapunov, teorema de Chetaev, solución de equilibrio, inestabilidad.

Resumen

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].

Biografía del autor

Cristian Cárcamo, Universidad del Bío-Bío.
Facultad de Ciencias, Departamento de Matemática.
Claudio Vidal, Universidad del Bío-Bío.
Facultad de Ciencias, Departamento de Matemática.

Citas

[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).
Publicado
2013-02-19
Cómo citar
Cárcamo, C., & Vidal, C. (2013). The Chetaev Theorem for Ordinary Difference Equations. Proyecciones. Journal of Mathematics, 31(4), 391-402. https://doi.org/10.4067/S0716-09172012000400007
Sección
Artículos