Explicit Solutions for a Class of Nonlinear PDE that Arise in Allocation Problems

Abstract

To exploit large deviation approximations for allocation and occupancy problems one must solve a deterministic optimal control problem (or equivalently, a calculus of variations problem). As this paper demonstrates, and in sharp contrast to the great majority of large deviation problems for processes with state dependence, for allocation problems one can construct more-or-less explicit solutions. Two classes of allocation problems are studied. The first class considers objects of a single type with a parameterized family of placement probabilities. The second class considers only equally likely placement probabilities, but allows for more than one type of object. In both cases, we identify the Hamilton-Jacobi-Bellman equation whose solution characterizes the minimal cost, explicitly construct solutions, and identify the minimizing trajectories. The explicit construction is possible because of the very tractable properties of the relative entropy function with respect to optimization.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
May 12, 2006
Accession Number
ADA458951

Entities

People

  • Jim X. Zhang
  • Paul Dupuis

Organizations

  • Brown University

Tags

Communities of Interest

  • Materials and Manufacturing Processes

DTIC Thesaurus Topics

  • Algebra
  • Applied Mathematics
  • Calculus
  • Calculus Of Variations
  • Differential Equations
  • Equations
  • Integrals
  • Mathematics
  • Monotone Functions
  • Nonlinear Algebraic Equations
  • Notation
  • Numbers
  • Partial Differential Equations
  • Probability
  • Probability Distributions
  • Random Variables
  • Statistics

Readers

  • Adaptive Control and Estimation with Uncertainty in Dynamic Systems.
  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Theoretical Analysis.