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On Uniform Exponential Stability of Self adjoint Evolution Family: By Weak Rolewicz Approach

Author Affiliations

  • 1Department of Mathematics University of Peshawar, Peshawar, PAKISTAN

Res. J. Recent Sci., Volume 4, Issue (2), Pages 68-71, February,2 (2015)

Abstract

In this article we prove that if P = {P(s,t)}s≥t≥0 is a self adjoint and strongly q-periodic continuous evolution family of bounded linear operators acting on a complex or real Hilbert space H then P is uniformly exponentially stable if for each unit vector x ∈ H the integral ∫n ϕ(<P(s,0)x,x>) ds is bounded, whereϕ :R := [0,∞) → R+ is a non-decreasing function such that ϕ(0) = 0 and ϕ(s) >0 for all s ∈ (0∞) .

References

  1. Buse C. and Rahmat G., Weak Rolewicz theorem in Hilbert spaces, Electronic journal of Differential Equations, 218, 1-10 (2012)
  2. Datko R., Extending a theorem of A. M. Liapunov to Hilbert space, Journal of Math. Anal, Appl., 32, 610-616 (1970)
  3. Pazy A., On the Applicability ofLyapunov's Theorem in Hilbert Space, SIAM J. Math. Anal, , 291-294 (1972)
  4. Rolewicz S., On uniform N-equistability, J. Math. Anal. Appl., 115, 434-441 (1986)
  5. Zabczyk J., Remarks on the control of discrete time distributed parameter systems, SIAM, J. Control, 12,731-735 (1974)
  6. Przyluski K.M., On a discrete time version of a problem of A. J. Pritchard and J. Zabcyzyk, Proc. Roy. Soc. Edinburgh, Sect. A, 101, 159-161 (1985)
  7. Zheng Q., The exponential stability and the perturbation problem of linear evolution systems in Banach spaces, J. Sichuan Univ., 25,401-411 (1988)
  8. Littman W., A generalization of a theorem of Datko and Pazy, Lecture Notes in control and Inform. Sci., 130, Springer-Verlag, Berlin, 318-323 (1989)
  9. Greiner G., Voight J. and Wol M., On the spectral bound of the generator of semi groups of positive operators, J. Operator Theory, 5(2), 245-256 (1981)
  10. Huang F., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Di. Eq, , 43-56 (1983)
  11. Jan van Neerven, Straub B., and Weis L., On the asymptotic behaviour of a semi group of linear operators, Indag. Math. (N.S.), 4(6), 453-476 (1995)
  12. Buse C. and Dragomir S.S., A Theorem of Rolewicz's type on Solid Function Spaces, Glasgow Mathematical Journal, 44, 125-135 (2002)
  13. Buse C. and Dragomir S.S., A Rolewicz's type Theorem. An evolution semi group approach, Electronic Journal Differential Equations, 45, 1-5 (2001)
  14. Storozhuk K.V., On the Rolewicz theorem for evolution operators, Proc. Amer. Math. Soc., 135, 6, 1861-1863 (2007)