In this paper, the reachable set estimation problem is investigated for Markovian jump neural networks (NNs) with time-varying delays and bounded peak disturbances. Our goal is to find a set as small as possible which bounds all the state trajectories of the NNs under zero initial conditions. In the framework of Lyapunov-Krasovskii theorem, a newly-found summation inequality combined with the reciprocally convex approach is used to bound the difference of the proposed Lyapunov functional. A new less conservative condition dependent on the upper bound, the lower bound and the delay range of the time delay is established to guarantee that the state trajectories are bounded within an ellipsoid-like set. Then the result is extended to the case with incomplete transition probabilities and a more general condition is derived. Finally, examples including a genetic regulatory network are given to demonstrate the usefulness and the effectiveness of the results obtained in this paper.

In this paper, the problem of energy-to-peak filtering problem is studied for a class of continuous-time semi-Markov jump systems with mismatched modes. We consider the scenario where the mode of the system is not always instantaneously accessible and mismatches the mode of the filter. To cope with this asynchronous phenomenon, we introduce a hidden semi-Markov model and use a stochastic process as a switching signal of the filter to observe the system modes. By Lyapunov function approach and linear matrix inequality method, sufficient conditions are developed to ensure the filtering error dynamics is stable with a prescribed energy-to-peak performance index. Two examples are given to illustrate the effectiveness of the proposed design technique.

The focus of this paper is to address the asynchronous control issue for a class of nonlinear Markov jump systems with parameter uncertainties. They are represented by the interval type-2 (IT2) polynomial fuzzy model. A hidden Markov model is utilized to describe the asynchronous behavior between the system mode and the controller mode in a quantitative manner. This is achieved by using a joint random process, which is presented in a concise format and encompasses both spontaneous and simultaneous jumps. Further to facilitate the design flexibility and low implementation burden, the structure of the asynchronous polynomial fuzzy controller is formed based on the imperfect premise matching scheme. Facing both mismatched modes and mismatched premise variables, a joint Markov chain based membership-function-dependent stability criteria is established by utilizing the Lyapunov-Krasovskii theory and the $\mathcal {H}_{\infty }$ performance analysis is then conducted using a sum of square approach. The polynomial feedback gain parameters can be readily solved by SOSTOOLS. Finally, a numerical example is used to validate the effectiveness of our obtained results and the application potential is verified by a single-link robot arm model.

This paper investigates the asynchronous eventtriggered control problem for nonlinear Markov jump systems with complex transition probabilities. The nonlinear dynamics are captured by a polynomial fuzzy model, with fewer fuzzy rules and stronger approximation capacity compared with T-S fuzzy counterpart. In view of engineering applications, system mode information is not fully identified, including the mode state and the mode transitions. Here a hidden Markov model is utilized to serve as a detector of the original system mode. The transition probability of the system and the associated conditional transition probability for the joint process are both assumed to be general, which contains unknown or inaccurately known values. An event-based mode-dependent fuzzy controller is then constructed such that the considered nonlinear Markov jump systems can be stabilized, for the purpose of reducing data transmission and power consumption. With the aid of a new decoupling strategy, feasible conditions are derived and included in sum of square based theorems. Further the concept of imperfectly premise matching scheme is introduced to facilitate the membership-function-dependent stability analysis, so as to inject richer information of the membership function and obtain less conservative results. The flexibility of controller design is thus endowed. Examples are given to show the effectiveness of our obtained results.