Optimal control of a population dynamics model with hysteresis

TL;DR

This paper develops an optimal control framework for a complex population dynamics model involving prey, predator, and vegetation, incorporating hysteresis effects and nonconvex control constraints, with results on existence and relaxation of solutions.

Contribution

It introduces a novel control approach for a nonlinear PDE system with hysteresis and nonconvex constraints, providing relaxation results and nearly optimal solutions.

Findings

Existence of nearly optimal solutions established.

Relaxation results for nonconvex control constraints.

Model incorporates hysteresis effects in population dynamics.

Abstract

This paper addresses a nonlinear partial differential control system arising in population dynamics. The system consist of three diffusion equations describing the evolutions of three biological species: prey, predator, and food for the prey or vegetation. The equation for the food density incorporates a hysteresis operator of generalized stop type accounting for underlying hysteresis effects occurring in the dynamical process. We study the problem of minimization of a given integral cost functional over solutions of the above system. The set-valued mapping defining the control constraint is state-dependent and its values are nonconvex as is the cost integrand as a function of the control variable. Some relaxation-type results for the minimization problem are obtained and the existence of a nearly optimal solution is established.