Optimal control for elliptic Volterra-Lotka type equations (Q2367614)

The paper considers a controlled elliptic partial differential equation of the form \(\Delta u+u [(a(x) - f(x)) - bu]=0\) in a domain \(\Omega\), with no flux boundary condition. Here \(u(x)\) is the concentration function, while \(f(x)\) is the control effort. The function \(a(x)\) is the analog of a carrying capacity and \(b\) reflects a crowding effect. A harvesting policy \(f(x)\) is optimal with respect to the steady state equation if it maximizes harvesting (namely, the integral of \(Ku(x)f(x)\) over \(\Omega)\) minus cost (namely integrating \(Mf^ 2(x)\) over \(\Omega)\). The paper provides conditions (primarily \(a(x)\), \(f(x)\) in \(L^ \infty_ +\) and \(a (\cdot)\) bounded away from 0) for the existence of unique positive solutions for the state equations, and existence of optimal solutions. A scheme for computing the solution under an additional condition is provided, and the scheme is demonstrated on an example.