In this paper, we propose the False-Alarm-Rate-Controllable (FAC) classification of sea clutter Recurrence Plots (RPs) based on Convolutional Neural Networks (CNN), which is shortened as RPs-CNN. Sea clutter data is a non-linear and recursive time series, and RPs provide qualitative analysis of non-linear and recursive dynamic systems. Thus, we construct the RPs datasets to extract the recursive feature of sea clutter. In the RPs datasets, RPs parameters, embedding delay τ and embedding dimension m, are obtained by the average mutual information (AMI) and false nearest neighbor (FNN) algorithms, respectively. In addition, in order to extract the local features of RPs, CNN is applied. CNN makes full use of local features of the datasets, has the advantages of translation invariance and strong generalization ability, and shares the available weights to simplify network structure. We implement a proper LeNet-5 CNN training on the constructed RPs datasets, and verity it by IPIX measured datasets. The experimental results demonstrate that the CNN successfully classifies the RPs of targets and clutter. Moreover, the feasibility of RPs-CNN is verified by seven aspects, i.e., Accuracy, Precision, False Alarm, Miss Rate, Recall, F1-measure and Kappa. In addition, six parameters that may affect the classification performance are also analyzed, including time-series length, embedding delay, embedding dimension, convolutional kernel size, kernel depth, and optimization function. The results indicate that the proposed RPs-CNN method can reach a 92.05% F1-measure and 87.19% Kappa which performs better than other classification methods. Meanwhile, the false alarm rate (FAR) of the RPs-CNN is testified by the FAC classification. Experimental results demonstrate that the proposed RPs-CNN significantly improves the detection probability over other classification detectors in low FAR cases.