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The constrained least absolute deviation (L1) estimator is an attractive alternative to both the unconstrained L1 estimator and the least-square estimator. This paper introduces a constrained L1 method and proposes two cooperative recurrent neural networks (CRNNs) for the constrained L1 estimator. Unlike existing cooperative neural networks, the proposed two CRNNs have a novel weighting cooperation scheme to integrate individual neural network information automatically. As a special case, the proposed continuous-time CRNN includes the existing continuous- time neural network for unconstrained L1 estimator. Compared with existing continuous-time neural networks for the constrained L1 estimator, the proposed continuous-time CRNN has a lower model complexity and the finite-time convergence to the exact optimal solution without any additional condition. Furthermore, compared with existing numerical algorithms for the constrained L1 estimator, in addition to a low computational complexity, the proposed two CRNNs are suitable for parallel implementation and can deal with the L1 estimation problem with degeneracy. The proposed two CRNNs are applied to parameter estimation problems under non-Gaussian noise environments. Simulation results demonstrate that the proposed CRNNs are indeed effective in dealing with the L1 estimation problem with nonunique solutions and in obtaining a better solution than relevant algorithms.