$\mathscr {L}_\infty$ Control for Positive Delay Systems With Semi-Markov Process and Application to a Communication Network Model

This paper deals with the problem of <inline-formula><tex-math notation="LaTeX">$\mathscr {L}_\infty$</tex-math> </inline-formula> control for positive delay systems with semi-Markov process. The system is subjected to a semi-Markov process that is time-varying, dependent on the sojourn time, and related to Weibull distribution. The main motivation for this paper is that the practical systems such as the communication network model (CNM) described by positive semi-Markov jump systems (S-MJSs) always need to consider the sudden change in the operating process. To deal with the corresponding problem, some criteria about stochastic stability and <inline-formula><tex-math notation="LaTeX"> $\mathscr {L}_\infty$</tex-math></inline-formula> boundedness are presented for the open-loop positive S-MJSs. Further, some necessary and sufficient conditions for state-feedback controller satisfying <inline-formula> <tex-math notation="LaTeX">$\mathscr {L}_\infty$</tex-math></inline-formula> boundedness and positivity of the resulting closed-loop system is established in standard linear programming. Finally, the practical system about the CNM is given to verify the validity of the proposed method.