Chaperoning State Evolutions by Variable Durations

Abstract

McKendrick partial differential equations in demography provide the state of a population depending on chronological time $T$ and age $D$, and involve two time evolutions: the chronological one $t \in [0,T] \mapsto t \geq 0$ and the calendar age $t \in [T-D,T] \mapsto t-(T-D) \geq 0 $, both with constant velocity equal to $1$. The calendar age evolution chaperons, so to speak, the evolution $t \in [T-D,T] \mapsto x(t)$ of the state of a system. Some physical, biological, and economic problems motivate the introduction of variable durations with variable velocities (representing the fluidity of time) offering mathematical metaphors of a “subjective fleeting specious time” passing more or less slowly. Variable durations are no longer prescribed, but chosen among those available and regulated: the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion (or control system) and provides, as a byproduct, the unknown temporal windows on which they evolve toge...