Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type

摘要

We present some of our previous results (1995, 1998) where we treat an optimal control problem for a nonlinear elliptic equation of the Volterra-Lotka type with homogeneous Dirichlet boundary conditions. This type of problems arises from population dynamics, where they model the steady-state solutions of the corresponding nonlinear evolution problem. We are interested in maximizing a functional which represents the difference between economic revenue and cost. Fist, we prove the existence of optimal control. Then, assuming that the quotient between the price of the species and the cost of the control is small, an optimality system is derived. This is used for proving the uniqueness and constructive approximation to the optimal control. In the proofs, we use different methods related to the theory of nonlinear elliptic equations and nonlinear analysis, such as upper and lower solutions notions, variational characterization of eigenvalues, weak maximum principles, strictly convex operators, etc.