Long time behavior for a fractional Picard problem in a Hilbert space
Of concern is a nonlinear second order initial value differential problem involving a convolution of a singular kernel with the derivative of the state. The problem describes the dynamics of a single-degree-of-freedom fractional oscillator. It is a generalization of the standard harmonic oscillator. The model also generalizes some well-known fractionally damped second order differential equations such as the Bagley–Torvik equation. Moreover, it extends models using exponential non-viscous damping to the more challenging singular case. We prove an exponential stability result of the equilibrium using the multiplier technique. A new energy functional, different from the classical one and different from the one obtained by the diffusive representation, is introduced.
KeywordsExponential decay Memory term Fractional damping Multiplier technique
AMS subject classifications34D20 34K37
The authors are thankful to the referees of the present paper for their careful reading and valuable comments. The second author is grateful to King Fahd University of Petroleum and Minerals for its continuous financial support through Project No. IN161010.
- 2.Biot, M. A.: Linear thermodynamics and the mechanics of solids. In: Proceedings of Third U. S. National Congress of Applied Mechanics. pp. 1–18 (1958)Google Scholar
- 20.Naber, M.: Linear fractionally damped oscillator. Intern. J. Diff. Equ. Article ID 197020, 12 pages (2010)Google Scholar
- 33.Wang, Z.H.: Solution and stability of a linear fractionally damped oscillator. In: Stepan, G., et al. (eds.) Dynamics Modeling & Interaction Control. IUTAM Book Series 30, pp. 101–108. Springer, Berlin (2011)Google Scholar