Michael McAsey

Convergence of the forward-backward sweep method in optimal control

The Forward-Backward Sweep Method is a numerical technique for solving optimal control problems. The technique is one of the indirect methods in which the differential equations from the Maximum Principle are numerically solved. After the method is briefly reviewed, two convergence theorems are proved for a basic type of optimal control problem. The first shows that recursively solving the system of differential equations will produce a sequence of iterates converging to the solution of the system. The second theorem shows that a discretized implementation of the continuous system also converges as the iteration and number of subintervals increases. The hypotheses of the theorem are a combination of basic Lipschitz conditions and the length of the interval of integration. An example illustrates the performance of the method.

An Application of Optimal Control to the Economics of Recycling

A citys landfill is an exhaustible resource; recycling is a backstop method of waste disposal. We formulate an optimal control model that maximizes the total utility of a representative consumer by choosing the appropriate levels of the two disposal techniques. The model is simple enough that some general conclusions can be drawn, and yet specific solutions will require some computations that show a few of the possibilities in optimal control problems. Among the primary results is that once recycling begins, it will increase. It is also likely that the level of recycling will assume the values at the endpoints of its domain as well as the more standard values in the interior.

Existence and Characterization of Optimal Locations

A general model for optimal location problems is given and the existence of solutions is proved under practical conditions. Conditions that all possible solutions must satisfy are given; these conditions form the basis of a method of finding solutions.

On projective equivalence of invariant subspace lattices

Abstract For i = 1,2, let A i be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A 1 onto the invariant subspace lattice of A 2 . We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds).

Properties of Derivates and Some Applications

In this chapter, we generalize the concept of derivates, defined recently in the literature, to maps defined on a topological space. The derivate of a map has some interesting properties and applications to optimization problems. For example, it is closely related to various notions of tangent spaces of the range of the map. It strengthens the necessary condition (Fermat’s theorem) for an extremum point to a sufficient condition.

Gains and losses from transfers of solid waste

Municipal solid waste is often transferred to landfills in other regions or states. While municipalities frequently resist imports, the interpretations of the Interstate Commerce Clause require that landfills accept waste regardless of its origin. This may require importers to act in ways that are not in their own best interest. The analysis of this paper suggests that importers benefit from trade when their landfill is so large that it is not exhausted at the end of the planning period. However, when landfill capacity is sufficiently scarce, importing waste is not welfare enhancing. Such municipalities have considerable motivation to try to de facto exclude waste from outside the municipality.

Locating a recycling center: The general density case

This paper considers a municipality that has a landfill (in a fixed location) and plans to optimally locate a recycling center to minimize transportation costs. The transportation problem consists of two stages. The first stage is the transportation of waste from households to the recycling center. Households are distributed (not necessarily uniformly) over the two-dimensional city. The second stage is the transportation of non-recyclables from the recycling center to the landfill. A precise description of the recycling centers optimal location depends on the density function, the proportion of recycled waste, and the location of the landfill.