Distributed-Order Non-Local Optimal Control

Abstract

Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

Optimal Control Problems Involving Combined Fractional Operators with General Analytic Kernels

“…If Ω(t) = L 2 , then N Ω(t) (u * ) = 0, and the optimality condition (13) is reduced to ∂H ∂u (t, x * (t), u * (t), λ(t)) = 0, which gives the particular result obtained in [19] (see also [27], in a different context).…”

Pontryagin Maximum Principle for Distributed-Order Fractional Systems

“…The results were then further generalized by the present authors in Reference [12], with the proof of several analytical results and a weak maximum principle of Pontryagin type for distributed-order fractional optimal control problems. Here, we extend and improve the theory of optimal control for distributed-order fractional operators initiated in Reference [12] by proving a strong version of the Pontryagin maximum principle, which allows the values of the controls to be constrained to a closed set. The main novelty consists to extend the optimality condition proved in Reference [12] to a maximality condition, which yields to the strong version of Pontryagin maximum principle.…”

“…The theory of the calculus of variations for distributed-order fractional systems was initiated in 2018 by Almeida and Morgado [11], and it has been extended by the authors in 2020 to the more general framework of optimal control [12]. There, we established a weak Pontryagin Maximum Principle (PMP), under certain smoothness assumptions on the space of admissible functions, where the controls are not subject to any pointwise constraint [12]. The objective of the present article was to state and prove a strong version of the PMP for distributed-order fractional systems, valid for general non-linear dynamics and L ∞ controls and, in contrast with References [11,12], without assuming that the controls take values on all the Euclidean space.…”

“…There, we established a weak Pontryagin Maximum Principle (PMP), under certain smoothness assumptions on the space of admissible functions, where the controls are not subject to any pointwise constraint [12]. The objective of the present article was to state and prove a strong version of the PMP for distributed-order fractional systems, valid for general non-linear dynamics and L ∞ controls and, in contrast with References [11,12], without assuming that the controls take values on all the Euclidean space. Our statement is as general as possible, and it encompasses the distributed-order calculus of variations of [11] and the weak PMP of [12] as particular cases.…”

“…Necessary optimality condition of Euler-Lagrange type for distributed-order arXiv:2108.03600v1 [math.OC] 8 Aug 2021 problems of the calculus of variations were first introduced and developed in Reference [11]. The results were then further generalized by the present authors in Reference [12], with the proof of several analytical results and a weak maximum principle of Pontryagin type for distributed-order fractional optimal control problems. Here, we extend and improve the theory of optimal control for distributed-order fractional operators initiated in Reference [12] by proving a strong version of the Pontryagin maximum principle, which allows the values of the controls to be constrained to a closed set.…”