Value functional and optimal feedback control in linear-quadratic optimal control problem for fractional-order system

TL;DR

This paper develops a method to compute optimal feedback controls for linear fractional-order systems by deriving a value functional and solving a fractional Hamilton-Jacobi-Bellman equation, enabling precise control strategies.

Contribution

It provides an explicit formula for the value functional and a step-by-step procedure for constructing epsilon-optimal controls in fractional-order systems.

Findings

Explicit formula for the value functional involving a Fredholm integral equation

A constructive feedback control procedure for epsilon-optimal controls

Analysis of the fractional Hamilton-Jacobi-Bellman equation with fractional derivatives

Abstract

In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing $\varepsilon$-optimal controls with any accuracy $\varepsilon > 0$ is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton-Jacobi-Bellman equation with so-called fractional coinvariant derivatives.