Optimal Control of State Constrained Age-Structured Problems

JOSEPH FRÉDÉRIC BONNANS; JUSTINA GIANATTI

Rio de Janeiro

Workshop; Workshop on Optimal Control and Mean Field Games; 2019

The age-structured systems appear naturally in many applications, in particular to model population dynamics, like it was introduced in [10, 9]. There are several works that study the existence, uniqueness, regularity and stability of solution for different age-structured systems with and without diffusion, for instance in [13, 12, 8, 5, 16]. The optimal control of age-structured systems has also been studied extensively, where optimality conditions and existence of optimal controls are obtained for certain cases as in [4, 7, 1, 2, 15]. However, in the case of optimal control problems with age-structured dynamics and state constraints, there are only a few works, such as [6, 11, 14].In this work, we address a general optimal control problem of partial differential equations, where the state equation is age-structured, and is driven by a general abstract parabolic operator. Most of the population dynamic models considered in the above mentioned works, are particular cases of our formulation. We also consider a finite number of linear state constraints. We start by proving the existence and uniqueness of solution of our system and we show that the first and second parabolic estimates hold. By analyzing the differentiability of the cost function and based on the general theory of Lagrange multipliers, we give a first order optimality condition. We define the costate of the system and we analyze its regularity by introducing an alternative costate, as in [3].