**1. Introduction**

Distributed-order fractional operators were introduced and studied by Caputo at the end of the previous century [1,2]. They can be seen as a kind of generalization of fractional order derivatives/integrals in the sense that these operators are defined by a weighted integral of different orders of differentiation over a certain range. This subject gained more interest at the beginning of the current century by researchers from different mathematical disciplines, through attempts to solve differential equations with distributed-order derivatives [3–6]. Moreover, at the same time, in the domain of applied mathematics, those distributed-order fractional operators have started to be used, in a satisfactory way, to describe some complex phenomena modeling real world problems—see, for instance, works in viscoelasticity [7,8] and in diffusion [9]. Today, the study of distributed-order systems with fractional derivatives is a hot subject—see, e.g., [10–12] and references therein.

Fractional optimal control deals with optimization problems involving fractional differential equations, as well as a performance index functional. One of the most important results is the Pontryagin Maximum Principle, which gives a first-order necessary optimality condition that every solution to the dynamic optimization problem must verify. By applying such a result, it is possible to find and identify candidate solutions to the optimal control problem. For the state of the art on fractional optimal control, we refer the readers to [13–15] and references therein. Recently, distributed-order fractional problems of the calculus of variations were introduced and

investigated in [16]. Here, our main aim is to extend the distributed-order fractional Euler–Lagrange equation of [16] to the Pontryagin setting (see Remark 2).

Regarding optimal control for problems with distributed-order fractional operators, the results are rare and reduce to the following two papers: [17,18]. Both works develop numerical methods while, in contrast, here we are interested in analytical results (not in numerical approaches). Moreover, our results are new and bring new insights. Indeed, in [17], the problem is considered with Riemann–Liouville distributed derivatives, while in our case we consider optimal control problems with Caputo distributed derivatives. We must also note an inconsistency in [17]: when one defines the control system with a Riemann–Liouville derivative, then in the adjoint system it should appear as a Caputo derivative—when one considers optimal control problems with a control system with Caputo derivatives, the adjoint equation should involve a Riemann–Liouville operator—as a consequence of integration by parts (cf. Lemma 1). This inconsistency has been corrected in [18], where optimal control problems with Caputo distributed derivatives (as in this paper) are considered. Unfortunately, there is still an inconsistency in the necessary optimality conditions of both [17,18]: the transversality conditions are written there exactly as in the classical case, with the multiplier vanishing at the end of the interval, while the correct condition, as we prove in our Theorem 1, should involve a distributed integral operator—see condition (3).

The text is organized as follows. We begin by recalling definitions and necessary results of the literature in Section 2 of preliminaries. Our original results are then given in Section 3. More precisely, we consider fractional optimal control problems where the dynamical system constraints depend on distributed-order fractional derivatives. We prove a weak version of Pontryagin's maximum principle for the considered distributed-order fractional problems (see Theorem 1) and investigate a Mangasarian-type sufficient optimality condition (see Theorem 2). An example, illustrating the usefulness of the obtained results, is given (see Examples 1 and 2). We end with Section 4 of conclusions, mentioning also some possibilities of future research.
