Technical comments on the optimal control for linear distributed time-delay

TL;DR

This paper addresses the infinite horizon optimal control problem for linear distributed time-delay systems, providing a formal justification for the Bellman functional's properties and demonstrating its positive definiteness, supported by experimental results.

Contribution

It introduces a novel approach using the Cauchy solution for distributed time-delay systems and formally justifies the Bellman functional's properties, including bounds and positive definiteness.

Findings

Bellman functional admits upper and lower bounds

Bellman functional is positive definite

Experimental results on temperature process support the theory

Abstract

The solution to the infinite horizon optimal control problem for linear distributed time-delay systems is presented. The proposal is based on the use of the Cauchy solution for distributed time-delay systems. In contrast with previous results (for punctual time delay systems), the form of the functional and its properties are formally justified. An important property is demonstrated: the Bellman functional admits upper and lower bounds, thus it is proved to be positive definite. Additionally, some experimental results on a temperature process are presented.