MAINTENANCE OF AGE-STRUCTURED POPULATIONS: OPTIMAL CONTROL, STATE CONSTRAINTS, AND BANG-BANG REGIME

The paper analyzes the optimal maintenance of a "fully manageable" biological population with no natural reproduction. Examples of such populations include dairy farms, horse breeding, gardening, and forestry. Mathematically, the problem leads to the non-linear optimal control of a first-order partial differential equation with pointwise state constraints. A necessary extremum condition is proven using the abstract optimization theory in Banach spaces. It is shown that the optimal dynamics has a bang-bang structure with three jumps and involves buying new young individuals and selling the old ones when their anticipated future expenses exceed the profit.