Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 70, pp. 1-12.
Title: Energy decay estimates for Lienard's equation with
quadratic viscous feedback
Authors: Alexander Y. Khapalov (Washington State Univ., Pullman, WA, USA)
Parthasarathi Nag (Washington State Univ., Pullman, WA, USA)
Abstract:
This article concerns the stabilization for a well-known
Lienard's system of ordinary differential equations modelling oscillatory
phenomena. It is known that such a system is asymptotically stable
when a linear viscous (motion-activated) damping with constant gain is
engaged. However, in many applications it seems more realistic that the
aforementioned gain is not constant and does depend on the deviation from
equilibrium. In this article, we consider a (nonlinear) gain, introduced in
[2], which is proportional to the square of such deviation and
derive an explicit energy decay estimate for solutions of the corresponding
``damped'' Lienard's system. We also discuss the place of our result in the
framework of stabilization of so-called critical bilinear systems.
Submitted April 9, 2003. Published June 21, 2003.
Math Subject Classifications: 93D05, 93D15, 93D20.
Key Words: Bilinear systems; stabilization; quadratic feedback;
energy decay estimate.