Explicit bounds on the exponential decay and on the L2 gain for linear time-varying systems

It is well known since many years ago that for the linear time-varying system e = Ae + B (t)θ, θ = -B(t)^Te with A Hurwitz, and B(t) bounded and globally Lipschitz, it is necessary and sufficient for global exponential stability, that B(t) satisfy the so-called persistency of excitation condition. In this note we provide explicit bounds for the convergence rate and the overshoot of the transient behavior of the solutions e(t), θ(t) as functions of the richness of B(t). Then we use the exponential decaying bound to obtain a converse Lyapunov function and finally, we provide an estimate on the L₂ gain of the system from an additive input. In particular we exhibit for this bound, its explicit dependence on the PE property of the regressor which is imposed as hypothesis.