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# Accession Number:

## AD0679231

# Title:

## A TRANSFORMATION TECHNIQUE FOR OPTIMAL CONTROL PROBLEMS WITH A STATE VARIABLE INEQUALITY CONSTRAINT.

# Descriptive Note:

## Technical rept.,

# Corporate Author:

## HARVARD UNIV CAMBRIDGE MASS DIV OF ENGINEERING AND APPLIED PHYSICS

# Report Date:

## 1968-10-01

# Pagination or Media Count:

##
29.0

# Abstract:

## A slack variable is used to transform an optimal control problem with a scalar control and a scalar inequality constraint on the state variables into an unconstrained problem of higher dimension. It is shown that, for a p-th order constraint, the p-th time derivative of the slack variable becomes the new control variable. The usual Pontryagin Principle of Lagrange multiplier rule gives necessary conditions of optimality. There are no discontinuities in the adjoint variables. A feature of the transformed problem is that any nominal control function produces a feasible trajectory. The optimal trajectory of the transformed problem exhibits singular arcs which correspond, in the original constrained problem, to arcs which lie along the constraint boundary this suggests a duality between singular and state-constrained problems, which should be explored. Generalizations of the approach to cases where the constraint and the control are vectors of equal dimension, as well as to problems involving multiple constraints and a single control variable, are considered. Owing to the appearance of singular arcs in the solution of the transformed problem, a direct application of second-order or second-variation algorithms is not possible. However, gradient or conjugate gradient methods are applicable and computations, using the conjugate gradient method, are presented to illustrate the usefulness of the transformation technique. Author

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#