ON A DISTRIBUTED CONTROL PROBLEM ARISING INDYNAMIC OPTIMIZATION OF A FIXED-SIZE POPULATION

摘要

The practical motivation for this paper is provided by the recruitment problem faced by many organizations of fixed size: to keep the average age young (and thus keep innovation awl productivity high) while at the same time keeping levels of recruitment high. A typical example is an academy of sciences. The problem is formalized by an infinite horizon optimal control model for a first order PDE with nonlocal dynamics (a McKendrick-type equation). Based on the nonstandard necessary optimality condition proved in the paper, the following results are established: (i) station-arity of the optimal recruitment density; (ii) strong ergodicity of the optimal solution; (iii) principle of "bipolar" recruitment in the case where the productivity of the organization is measured by the average age of the members. The analysis involves a new type of transversality condition for the costate system, the stability with respect to perturbations of the optimal solution of a noncoercive (bang-bang type) problem, boundedness, and stability of the solution of a specific Volterra integral equation of the second kind.