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This study is focused on analyzing the L 1 -stochastic internal stability and the design of an L 1 -gain controller for two-dimensional (2D) continuous positive Markov jumping systems (PMJSs) with constraints on the inputs and states. An algorithm is developed to explicitly determine the state feedback control law with the optional L 1 -gain performance. First, by constructing a co-positive stochastic Lyapunov function for the positive system and establishing the equation for the Markov states' mathematical expectation, several sufficient and necessary conditions for L 1 -stochastic internal stability are derived. This analysis indicates that 2D PMJSs with directional delays are influenced by sizes of the system matrices and magnitude of delays, and transition matrix. Second, a method for calculating the L 1 -gain is formulated utilizing linear programming (LP), thus, an L 1 -gain controller is designed to guarantee that the closed-loop system is positive and L 1 -stochastically internally stable, while achieving optimal L 1 -gain performance. Two numerical and practical examples are considered, and the influence of delays and transition probability matrices on the L 1 -gain is specified to validate the preceding theoretical findings. [ABSTRACT FROM AUTHOR]

• The syntactic structure of natural language is import for the model to understand the sentences comprehensively. • The relative syntactic distance can be used to describe the degree of dependency between adjacent words in a sentence. • A relative syntactic distance-based language model RSD-LSTM is proposed to integrate syntactic information into LSTM. • Low-level and high-level syntactic structure features can be fused to generate hidden representations. • The experimental results of RSD-LSTM outperform existing state-of-the-art models on different tasks. In natural language, the relationship among the constituents of a sentence is usually tree-like: words, phrases, and clauses constitute a sentence hierarchically, and the dependency between different constituents induces the syntactic structure. Such a complex tree-like structure is vital for understanding natural languages. However, recurrent neural networks (RNNs) model languages sequentially and fail to encode a hierarchical syntactic dependency comprehensively, therefore causing the networks to underperform on comprehension-based tasks. In this paper, we propose a novel neural language model, called relative syntactic distance LSTM (RSD-LSTM), to capture the syntactic structure dependency dynamically. RSD-LSTM employs a convolutional neural network to compute the relative syntactic distance between sentences to represent the degree of dependency between words and modifies the gating mechanism of LSTM through the relative syntactic distance. Furthermore, we add a direct connection between hidden states to fuse high-level and low-level syntactic features. We conducted extensive experiments on language modeling. The results suggest that RSD-LSTM achieves improvements of 1.82 and 2.03 in perplexity compared with current top methods on the Penn Treebank and WikiText-2 datasets, respectively. Moreover, we conducted experiments on a machine translation application task. Experimental results of this task also show significant improvements of RSD-LSTM compared with baseline models. [ABSTRACT FROM AUTHOR]