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This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.

This paper proposes a non-smooth human influenza model with logistic source to describe the impact on media coverage and quarantine of susceptible populations of the human influenza transmission process. First, we choose two thresholds $ I_{T} $ and $ S_{T} $ as a broken line control strategy: Once the number of infected people exceeds $ I_{T} $, the media influence comes into play, and when the number of susceptible individuals is greater than $ S_{T} $, the control by quarantine of susceptible individuals is open. Furthermore, by choosing different thresholds $ I_{T} $ and $ S_{T} $ and using Filippov theory, we study the dynamic behavior of the Filippov model with respect to all possible equilibria. It is shown that the Filippov system tends to the pseudo-equilibrium on sliding mode domain or one endemic equilibrium or bistability endemic equilibria under some conditions. The regular/virtulal equilibrium bifurcations are also given. Lastly, numerical simulation results show that choosing appropriate threshold values can prevent the outbreak of influenza, which implies media coverage and quarantine of susceptible individuals can effectively restrain the transmission of influenza. The non-smooth system with logistic source can provide some new insights for the prevention and control of human influenza.

Microgrid is an effective way to integrate the distributed energy resources into the utility networks. One of the most important issues is the power flow control of grid-connected voltage-source inverter in microgrid. In this paper, the small-signal model of the power flow control for the grid-connected inverter is established, from which it can be observed that the conventional power flow control may suffer from the poor damping and slow transient response. While the new power flow control can mitigate these problems without affecting the steady-state power flow regulation. Results of continuous-domain simulations in MATLAB and digital control experiments based on a 32-bit fixed-point TMS320F2812 DSP are in good agreement, which verify the small signal model analysis and effectiveness of the proposed method.