Stability equivalence between regime-switching jump diffusion delayed systems and corresponding systems with piecewise continuous arguments and application to discrete-time feedback control

In this paper, we mainly study the equivalence of exponential stability for regime-switching jump diffusion delayed systems (RSJDDSs) and RSJDDSs with piecewise continuous arguments (RSJDDSs-PCA). Our results show that if one of the RSJDDS and the RSJDDS-PCA is $p$th moment exponentially stable, then another system is also $p$th moment exponentially stable when time delay and segment step size have a common upper bound, while both equations are almost surely exponentially stable, and we also provided a method to calculate this upper bound. In addition, as an application of the stability equivalence theorem, we design discrete-time state and mode observations feedback control to stabilize unstable RSJDDSs and investigate that controllers of the drift, diffusion, and jump terms are all able to play a stabilizing effect on the controlled system.