# Population planning; a distributed time optimal control problem

## Abstract

The time evolution of the age profile of a group of people, for instance the population of a certain country, can be described by a first-order partial differential equation. A time optimal control problem arises when the population must be brought from a given age profile to another desired one as quickly as possible. The birth rate, i.e. the number of births per unit of time, is the control variable and it serves as a boundary condition for the partial differential equation. To prevent the age distribution to become undesirable from an economical point of view during the transient to the final situation we require the working population to exceed a given fraction of the total population at each instant of time. This introduces a state constraint to the problem.

For the cases considered the following facts turn out. a) If age and time are discretized properly, a linear programming problem results, the solution of which equals an optimal solution of the continuous version of the problem. b) The time optimal control is not necessarily unique. A complete characterization of the class of all optimal controls can be given. c) Under certain conditions the class of all optimal controls contains a unique non-increasing control.

Two examples are solved analytically.

### 7. References

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