Dynamic Programming Principle and Hamilton-jacobi-bellman Equations for Fractional-order Systems

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. © 2020 Society for Industrial and Applied Mathematics.