Libin Mou

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.

Removability of singular sets of harmonic maps

It is proved that a harmonic map with small energy and the monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density; moreover, the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability. This work generalizes the results in [CL, DF, LG12], which were proved under the assumption that the singular sets are isolated points or smooth submanifolds.

Regularity forn-harmonic maps

Here we obtain everywhere regularity of weak solutions of some nonlinear elliptic systems with borderline growth, includingn-harmonic maps between manifolds or map with constant volumes. Other results in this paper include regularity up to the boundary and a removability theorem for isolated singularities.

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in Rm,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou ¦ϖΩ is Fredholm of index 0.

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.

Existence of Solutions of a Riccati Differential System from a General Cumulant Control Problem

We study a system of infinitely many Riccati equations that arise from a cumulant control problem, which is a generalization of regulator problems, risk-sensitive controls, minimal cost variance controls, and k-cumulant controls. We obtain estimates for the existence intervals of solutions of the system. In particular, new existence conditions are derived for solutions on the horizon of the cumulant control problem.

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.

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.

Harmonic Maps for Bumpy Metrics

Here we give a description on how a harmonic map (from one manifold to another) varies, depending on the deformation of the image manifold. We are particularly interested in the deformations which keep the singularities of harmonic maps. An application is given.