Bulletin of the Australian Mathematical Society

Research Article

CONVERGENCE OF SOLUTIONS OF TIME-VARYING LINEAR SYSTEMS WITH INTEGRABLE FORCING TERM

JITSURO SUGIEa1

a1 Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan (email: jsugie@riko.shimane-u.ac.jp)

Abstract

The following system is considered in this paper:

\[ x' = -e(t)x+f(t)y, \quad y'=-g(t)x-h(t)y+p(t). \]

The primary goal is to establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) and a forcing term p(t) for all solutions to converge to the origin (0,0) as $t \to \infty $. Here, the zero solution of the corresponding homogeneous linear system is assumed to be neither uniformly stable nor uniformly attractive. Sufficient conditions are given for asymptotic stability of the zero solution of the nonlinear perturbed system
\[ x' = -e(t)x+f(t)y, \quad y'=-g(t)x-h(t)y+q(t,x,y) \]

under the assumption that q(t,0,0)=0.

(Received February 27 2008)

2000 Mathematics subject classification

  • 34D05;
  • 34D10;
  • 34D20

Keywords and phrases

  • nonhomogeneous linear systems;
  • weakly integrally positive;
  • perturbation problems

Footnotes

Supported in part by Grant-in-Aid for Scientific Research, No. 19540182.