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# Accession Number:

## AD0475436

# Title:

## ON THE GEOMETRY OF OPTIMAL PROCESSES. PART II,

# Descriptive Note:

# Corporate Author:

## CALIFORNIA UNIV BERKELEY INST OF ENGINEERING RESEARCH

# Report Date:

## 1965-11-01

# Pagination or Media Count:

##
117.0

# Abstract:

## This report contains an investigation of the geometry in state space of a dynamic system which behaves in an optimal fashion. The general notion of a dynamical system is introduced in terms of a set of admissible rules which determine the motion of the system in its state space. A cost is associated with the transfer of the system by means of an admissible rule. Optimality is defined by the requirement that an admissible rule render the minimum value of the cost associated with transfer between prescribed end states. Under the sole assumption that the cost obeys an additivity property, the existence of so-called limiting surfaces in cost-augmented state space is exhibited. Each member of the one-parameter family of limiting surface is the locus of all optimal trajectories whose initial points belong to it. Under various assumptions concerning the geometry of limiting surfaces, local properties of these surfaces are deduced. While it is not the primary purpose of the investigation reported to present a derivation of the Maximum Principle, this principle is found to be a consequence of the global and local perperties of limiting surfaces. Finally, the relation between the Maximum Principle and Dynamic Programming is established from the geometric point of view. Author

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#