Explicit bounds on the exponential decay and on the L/sub 2/ 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)/spl theta/, /spl theta/ = -B(t)/sup T/e 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), /spl theta/(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 L/sub 2/ 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.