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# Accession Number:

## ADA188086

# Title:

## Viscosity Methods in Optimal Control of Distributed Systems.

# Descriptive Note:

## Annual technical rept. 1 Aug 86-15 Aug 87,

# Corporate Author:

## LOYOLA UNIV OF CHICAGO IL DEPT OF MATHEMATICAL SCIENCES

# Report Date:

## 1987-08-15

# Pagination or Media Count:

##
4.0

# Abstract:

## The rigorous connection between viscosity solutions for the Bellman equation in optimal control and the Pontryagin Maximum Principle has been established. The method developed for controlled ordinary differential equations was extended to infinite dimensions to derive the Pontryagin principle for 1 a class of controlled nonlinear evolution equations in a Hilbert space, 2 a class of controlled nonlinear, divergence form parabolic partial differential equations and 3 a class of differential-difference equations. Additional subjects were studied were the extension of the idea of viscosity solution to equations with only time-measureable Hamiltonians and the optimal cooling of a free boundary problem with Stefan problem dynamics. Two problems of interest in specific applications were solved. The optimal control is characterized in the class of monotone functions which minimizes the H1 distance to a given function. This problem, a specific monotone follower problem, arises in production planning. An optimal portfolio selection problem is considered which includes stock, options, bonds and borrowed cash at an interest rate different from the bond interest rate. This problem is formulated using stochastic optimal control and explicitly constructed the solution of the Bellman equation. The objective of the study was to derive the option price which the market sets to minimize the investors maximal expected utility of wealth.

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#