*Article* **Distributed-Order Non-Local Optimal Control**

#### **Faïçal Ndaïrou †,‡ and Delfim F. M. Torres \*,‡**

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal; faical@ua.pt


Received: 9 September 2020; Accepted: 22 October; Published: 25 October 2020

**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.

**Keywords:** distributed-order fractional calculus; basic optimal control problem; Pontryagin extremals

**MSC:** 26A33; 49K15
