On preservation of dissipation inequalities under sampling

We show that if we first design a controller for a continuous-time nonlinear plant with disturbances so that it achieves a certain dissipation inequality for the continuous-time closed-loop system and then implement it as a sampled-data controller using a sampler and zero order hold, then the dissipation inequality will be preserved for the exact discrete-time model of the sampled data closed-loop system in a semiglobal practical sense (the sampling period is the parameter that we can adjust). Moreover, a similar statement is proved for open-loop systems, where controls are considered as free variables. Two different forms of dissipation inequalities are considered for the exact discrete-time models: the "weak" form and the "strong" form.