On the convergence rate of ordinal comparisons of random variables

y 00 (except at t = 0). One additional differentiation to get y 000 guarantees full column rank equal to three for the coefficient matrix of vector x. For the example, this rank condition is equivalent to the 1-full condition on the Jacobian O j; k of (11) for the time-varying linearization. (We may take j = 1 and k = 3.) Thus we have smooth observability on the interval [0; T ] for the time-varying linearized system. Then, provided A6) holds (perhaps after additional differentiation, see Theorem 3), Theorem 2 applies and guarantees smooth observability of the non-linear system on [0; T ] in a neighborhood of x = 0. Based on Theorem 2, many special results may now be deduced, involving conditions on a linearization (8), (9), or on submatrices of the full Jacobian (7), that guarantee smooth observability in W about a trajectory, in particular for semi-explicit systems where (7) simplifies somewhat. For example, the analysis in [13] shows that the hypotheses in Theorem 2 will hold for Hessenberg DAE systems supplied with an effective output function, that is, an output for which O j; k in (11) is 1-full with respect to x on I. Finally, the special u-dependence dealt with in (1) is not a real restriction: it is still possible to apply the condition (6) to (7), allowing Theorems 2 and 3 of Section IV to be extended to the case of a DAE with the more general u-dependence, F (x; x 0 ; t; u) = 0, in place of (1). The linearization is still given by (8), (9), but in this case the Jaco-bian (7) also depends on the variables u (i) for 0 i j. V. CONCLUSION We have established some sufficient conditions for local observ-ability of nonlinear DAE systems near a known trajectory. We indicated by an example the importance of these results in overcoming the inadequacy of time-invariant linearizations for determining observability of nonlinear DAEs. Our sufficient conditions for smooth observability are verifiable and are strong enough to guarantee that the full system observability Jacobian satisfies the smooth observability condition in [13]. The results of this note can provide a basis for the future development of algorithms for determining observability of nonlinear DAE systems. ACKNOWLEDGMENT The author would like to thank a reviewer for helpful comments that resulted in a clearer note. Feedback stabilization of control systems described by …