Global Dynamics of Piecewise Smooth Systems with Switches Depending on Both Discrete Times and Status

This paper is concerned with global dynamics, especially periodicity, for a piecewise smooth system incorporating a new control strategy. The switches take place at discrete times and depend on the status. By employing the approach of Poincare'\ maps, the existence, exact number, and asymptotical stability of periodic solutions are investigated thoroughly in some parameter regions. The periodic solutions are induced by the control strategy. As applications, convergence and periodicity are studied both for a fishery model and for an SIS model with discrete time on-off control. Numerical simulations are performed to verify our results.