Hotelling Conservation

Chapter 3 explores the Hotelling theory of conservation. It shows how the Hotelling arbitrage condition provides a test for whether it is optimal to convert or conserve a resource in some state. If the value of a resource when conserved is expected to grow faster than its value when converted, it will be optimal to conserve it. The chapter shows how the arbitrage condition applies to both nonrenewable and renewable resources, and how it is embodied in the conditions required for the optimal conversion/conservation of natural resources of both kinds. It also introduces Hamiltonian methods commonly used to optimize natural resource management, and shows the relation between Lagrangian and Hamiltonian methods. Finally, it discusses the relation between Hotelling conservation and the methods applied in conservation biology.

Singular extremals in L1-optimal control problems: sufficient optimality conditions

In this paper we are concerned with generalised L1-minimisation problems, i.e. Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proved by means of Hamiltonian methods. As a by-product of the result, we provide an explicit invariant formula for the second variation along the singular arc.

Enumerative numerical solution for optimal control using treatment and vaccination for an SIS epidemic model

Optimal control problems in mathematical epidemiology are often solved by Hamiltonian methods. However, these methods require conditions on the problem to guarantee that they give global solutions. Because of the improved computational power of modern computers, numerical approximate solutions that systematically try a large number of possibilities have become practical. In this paper we give an efficientimplementation of an enumerative numerical solution method for an optimal control problem, which applies to cases where standard methods cannot guarantee global optimality. We demonstrate the method on a model where vaccination and treatment are used to control the level of prevalence of an infectious disease. We describe the solution algorithm in detail, and verify the method with simulations. We verify that the enumerative numerical method produces solutions that are locallyoptimal.