Lecture notes - ODES with Differentiated Inputs Specifying Preinitial Conditions

When a differential equation has differentiated inputs, the pre-initial values of the input must be specified in order for the problem to be well posed. These values are not determined by the mathematics but rather by the physical situation. The issue persists in state-space models, even though no derivatives of the input appear explicitly. Consequently, care must be taken to determine the pre-initial conditions to obtain the physically meaningful solution. The purpose of this paper is to stress the critical role of the initial conditions of the input when the differential equation involves differentiated inputs.