The quadratic regulator problem and the Riccati equation for a process governed by a linear Volterra integrodifferential equations

TL;DR

This paper establishes a connection between the quadratic regulator problem for systems governed by linear Volterra integrodifferential equations and a Riccati differential equation, enabling feedback control solutions.

Contribution

It introduces a novel Riccati differential equation framework for Volterra systems, extending control theory beyond previous Fredholm integral equation approaches.

Findings

Derivation of a Riccati differential equation for Volterra systems

Feedback form of optimal control for the studied process

Extension of control theory to integrodifferential equations

Abstract

In this paper we study the quadratic regulator problem for a process governed by a Volterra integral equation in ${\mathbb R}^n$. Our main goal is the proof that it is possible to associate a Riccati differential equation to this quadratic control problem, which leads to the feedback form of the optimal control. This is in contrast with previous papers on the subject, which confine themselves to study the Fredholm integral equation which is solved by the optimal control.