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 α ∈ (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 α, 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.

Optimal Control Problems with a Fixed Terminal Time in Linear Fractional-Order Systems

“…According to [4], by a position of system (1), we mean a pair (t, w(•)) consisting of a time t ∈ [t 0 , ϑ] and a function w(•) ∈ AC α ([t 0 , t], R n ), w(t 0 ) ≤ R x , which is treated as a history of a motion of the system on the time interval [t 0 , t]. The set of all such positions is denoted by G. Respectively, for every x 0 ∈ B(R x ), the pair (t 0 , x 0 ) ∈ G is regarded as an initial position.…”

“…We consider a formalization of feedback controls within the framework of positional control strategies [1,2] (see also [3,4]). By a (positional) control strategy, we mean an arbitrary function…”

“…One of them evaluates the state vector of the system realized at a fixed terminal time ϑ, and the other is an integral evaluation of a control on the whole time interval [t 0 , ϑ]. We are interested in finding optimal, or, at least, ε-optimal, open-loop controls, as well as in constructing optimal feedback (closed-loop) controls, which are formalized within the framework of positional strategies [1,2] (see also [3,4]).…”

“…In order to solve the problem, we propose an approach based on its reduction to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation and further application of the methods and results of the optimal control theory widely developed for such systems. The reduction relies on a suitable notion of a finitedimensional informational image of an infinite-dimensional position [4] of the original fractional-order system, which can be treated as a special prediction of a motion of this system at the time ϑ.…”