The Bilinear, Modal State Equations for Age-Dependent Growth Control

Abstract

There have been previous attempts to model biological processes as bilinear systems [4,9,10]. In these early studies any member of a population was taken to be quite like any other so that the variation of fertility and susceptibility to mortality with age was ignored. In this paper, however, the age-dependent nature of biological growth [5] is accounted for. The modal (eigenfunction) analysis of the basic partial differential equation of age-dependent growth is shown to result in a system of bilinear equations. (The basic mathematical model is a non-self-adjoint operator with a discrete spectrum and the modes are coupled by the control term.) The impulse control of a truncated version of this system of equations is then discussed. It is anticipated that the results presented here will aid planning for optimal amounts of pesticides to agro-ecosystems or for optimal amounts of drugs (or radiations) to unwanted cell populations.