The problem of finite/fixed-time cooperative state estimation is considered for a class of quasilinear systems with nonlinearities satisfying a H\"older condition. A strongly connected nonlinear distributed observer is designed under the assumption of global observability. By proper parameter tuning with linear matrix inequalities, the observer error equation possesses finite/fixed-time stability in the perturbation-free case and input-to-state stability with respect to bounded perturbations. Numerical simulations are performed to validate this design.

The universal approximation theorems [4], [10], [9] put limits on what artificial neural networks (ANNs) can theoretically learn. These theorems guarantee the existence of ANN, which approximates a continuous function on a compact set with an arbitrary high precision. A training of the ANN is based a compactly supported data as well, while, the trained ANN may be utilized next as a predictor of the function value for an input data, which does not belong to the training set. Sometimes, the new input may be have a rather large distance even from a convex hull of the training set. In the latter case, the ANN is utilized as an extrapolator of the function and the approximation theorems are not applicable. Moreover, the analysis of the extrapolation error is impossible if there is no information about the function away from the training set. Therefore, a global extrapolation of a function based on a local data can be provided only under additional assumption about the class of functions approximated by ANN. This paper deals with approximation of the so-called generalized homogeneous functions [24], [12], [2], [21] and introduces the corresponding homogeneous artificial neural network, which key feature is a global approximation based on local data. The generalized homogeneity is a symmetry of an object (a function, a set, a vector field, etc) with respect to a group of the so-called generalized dilations.