# Dynamic programming principle and Hamilton-Jacobi-Bellman equations for fractional-order systems

## TL;DR

This paper develops a dynamic programming framework for fractional-order control systems, establishing a Hamilton-Jacobi-Bellman equation and methods for constructing optimal feedback controls.

## Contribution

It introduces a new notion of coinvariant derivatives for fractional systems and links the value functional to a HJB equation under smoothness assumptions.

## Key findings

- Proves the dynamic programming principle for fractional systems
- Establishes a connection between the value functional and HJB equation
- Proposes a method for constructing optimal feedback controls

## Abstract

We consider a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1)$. The value of this problem is introduced as a functional in a suitable space of histories of motions. We prove that this functional satisfies the dynamic programming principle. Based on a new notion of coinvariant derivatives of the order $\alpha$, we associate the considered optimal control problem with a Hamilton-Jacobi-Bellman equation. Under certain smoothness assumptions, we establish a connection between the value functional and a solution to this equation. Moreover, we propose a way of constructing optimal feedback controls. The paper concludes with an example.