Abstract.
Let be a time-varying vector field depending on t containing a regular and a slow time scale (α large). Assume there exist a k (τ)≥1 and a γ(τ) such that ∥x τ(t, t 0, x 0)∥≤k(τ) e −γ(τ)(t−t0)∥x 0∥, with x τ(t, t 0, x 0) the solution of the parametrized system with initial state x 0 at t 0. We show that for α sufficiently large is exponentially stable when “on average”γ(τ) is positive. The use of this result is illustrated by means of two examples. First, we extend the circle criterion. Second, exponential stability for a pendulum with a nonlinear slowly time-varying friction attaining positive and negative values is discussed.
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Date received: January 22, 2000. Date revised: April 14, 2001.
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Peuteman, J., Peuteman, J. & Aeyels, D. Exponential Stability of Slowly Time-Varying Nonlinear Systems. Math. Control Signals Systems 15, 202–228 (2002). https://doi.org/10.1007/s004980200008
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DOI: https://doi.org/10.1007/s004980200008