Gronwall-Bellman type integral inequalities and applications to global uniform asymptotic stability

• Mekki Hammi ^[1] ; Mohamed Ali Hammami ^[1]
  1. [1] University of Sfax

     University of Sfax

     Túnez

     
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 17, Nº. 3, 2015, págs. 53-70
• Idioma: inglés
• DOI: 10.4067/S0719-06462015000300004
• Enlaces
  • Texto completo
• Resumen
  • español

    En este artículo, establecemos algunas desigualdades integrales nolineales nuevas de tipo Gronwall-Bellman. Estas desigualdades pueden ser usadas como herramientas utiles para estudiar problemas de estabilidad de sistemas dinámicos perturbados. Como aplicaciones, basados en las nuevas desigualdades establecidas, también damos algunos resultados nuevos de estabilidad uniforme prácticos. Un ejemplo numérico es presentado para ilustrar la validez de los resultados principales.

  • English

    In this paper, some new nonlinear generalized Gronwall-Bellman-Type integral inequalities are established. These inequalities can be used as handy tools to research stability problems of perturbed dynamic systems. As applications, based on these new established inequalities, some new results of practical uniform stability are also given. A numerical example is presented to illustrate the validity of the main results.

• Referencias bibliográficas
  • Bellman, R. (1953). Stability Theory of Differential Equations. McGraw Hill. New York.
  • Benabdallah, A,Dlala, M,Hammami, M.A. (2007). A new lyapunov function for stability of timevarying nonlinear perturbed systems. Systems and...
  • Ben Abdallah, A,Ellouze, I,Hammami, M.A. (2009). Practical stability of nonlinear time-varying cascade systems. Journal of Dynamical and Control...
  • Chaillet, A,Loria, A. (2006). Necessary and sufficient conditions for uniform semiglobal practical asymptotic stability: Application to cascaded...
  • Corless, M. (1900). Guaranteed rates of exponential convergence for uncertain systems. J. Optim. Theory Appl. 64. 481-494
  • Gronwall, T.H. (1919). Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math....
  • Hammami, M.A. (2001). On the stability of nonlinear control systems with uncertainly. J. Dynamical Control systems. 7. 171-179
  • Khalil, H.K. (2002). Nonlinear Systems. Prentice-Hall. New York.
  • Lin, Y,Sontag, D,Wang, Y. (1996). A smouth converse Lyapunov theorem for robust stability. SIAM Journal on control and optimisation. 34. 124-160
  • Malisoff, M. (2005). Further results on Lyapunov functions and domains of attraction for perturbed asymptotically stable systems. Dynamics...
  • Mu, X,Cheng, D. (2005). On Stability and Stabilization of Time-varying Nonlinear Control Systems. Asian J. Control. 7. 244-255
  • Panteley, E,Loria, A. (1998). On global uniform asymptotic stability of nonlinear time-varying systems in cascade. Systems Control Letters....
  • Panteley, E,Loria, A. (2001). Growth rate conditions for uniform asymptotic stability of cascade time-varying systems. Automatica. 37. 453-460
  • Vidyasagar, M. (1993). Nonlinear Systems Analysis. 2. Practice Hall.
  • Zubov, V. I. (1964). Methods of A.M. Lyapunov and their Application. P. Noordhoff. Groningen.