On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models

This paper formulates the properties of point reachability and approximate point reachability of either a targeted state or output values in a general dynamic system which possess a linear time-varying dynamics with respect to a given reference nominal one and, eventually, an unknown structured nonlinear dynamics. Such a dynamics is upper-bounded by a function of the state and input. The results are obtained for the case when the time-invariant nominal dynamics is perfectly known while its time-varying deviations together with the nonlinear dynamics are not precisely known and also for the case when only the nonlinear dynamics is not precisely known. Either the controllability gramian of the nominal linearized system with constant linear parameterization or that of the current linearized system (which includes the time-varying linear dynamics) are assumed to be non-singular. Also, some further results are obtained for the case when the control input is eventually saturated and for the case when the controllability gramians of the linear parts are singular. Examples of the derived theoretical results for some epidemic models are also discussed.