Acta Mathematica

, Volume 170, Issue 2, pp 275–307 | Cite as

Precise damping conditions for global asymptotic stability for nonlinear second order systems

  • Patrizia Pucci
  • James Serrin


Asymptotic Stability Order System Global Asymptotic Stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Artstein, Z. &Infante, E. F., On the asymptotic stability oscillators with unbounded damping.Quart. Appl. Math., 34 (1976), 195–199.MathSciNetGoogle Scholar
  2. [2]
    Ballieu, R. J. &Peiffer, K., Attractivity of the origin for the equation\(\ddot x + f\left( {t,x,\dot x} \right)\left| {\dot x} \right|^\alpha \dot x + g\left( x \right) = 0\) J. Math. Anal. Appl., 65 (1978), 321–332.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Burton, T. A., On the equationu″+f(x)h(u′)u′+g(u)=e(t).Ann. Mat. Pura Appl., 85 (1970), 227–285.MathSciNetGoogle Scholar
  4. [4]
    Leoni, G., Manfredini, M. &Pucci, P., Stability properties for solutions of general Euler-Lagrange systems.Differential Integral Equations, 5 (1992), 537–552.MathSciNetGoogle Scholar
  5. [5]
    Levin, J. J. &Nohel, J. A., Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics.Archive Rational Mech. Anal. 5 (1960), 194–211.MathSciNetGoogle Scholar
  6. [6]
    Nakao, M., Asymptotic stability for some nonlinear evolution equations of second order with unbounded dissipative terms.J. Differential Equations, 30 (1978), 54–63.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    Pucci, P. &Serrin, J., A general variational identity.Indiana Univ. Math. J., 35 (1986) 681–703.CrossRefMathSciNetGoogle Scholar
  8. [8]
    —, Continuation and limit properties for solutions of strongly nonlinear second order differential equations.Asymptotic Anal., 4 (1991), 97–160.MathSciNetGoogle Scholar
  9. [9]
    —, Global asymptotic stability for strongly nonlinear second order systems, inProc. Conf. on Nonlinear Diffusion Equations and Their Equilibrium States (N. G. Lloyd, W.-M. Ni, L. A. Peletier and J. Serrin, eds.), pp. 437–449. Birkhäuser, Boston-Basel-Berlin, 1992.Google Scholar
  10. [10]
    Pucci, P. & Serrin, J., Continuation and limit behavior for damped quasi-variational systems, inProc. Conf. on Degenerate Diffusions, 1992 (W.-M. Ni, L. A. Peletier and J. L. Vazquez, eds.). The IMA Volumes and its Applications, 47. To appear.Google Scholar
  11. [11]
    Salvadori, L., Famiglie ad un parametro di funzioni di Liapunov nello studio della stabilità.Symposia Math., 6 (1971), 309–330. Istituto Nazionale di Alta Matematica, Roma.MATHMathSciNetGoogle Scholar
  12. [12]
    Smith, R. A., Asymptotic stability ofx″+a(t)x′+x=0.Quart. J. Math. Oxford, 12 (1961), 123–126.MATHGoogle Scholar
  13. [13]
    Thurston, L. H. &Wong, J. S. W., On global asymptotic stability of certain second order differential equations with integrable forcing terms.SIAM J. Appl. Math., 24 (1973), 50–61.CrossRefMathSciNetGoogle Scholar

Copyright information

© Almqvist & Wiksell 1993

Authors and Affiliations

  • Patrizia Pucci
    • 1
  • James Serrin
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PerugiaperugiaItaly
  2. 2.Department of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations