The $l$-moment problem and optimal control for systems modeled by fractional equations with multiparameter and “nonsingular” derivatives

This paper devoted to the problem of non-uniqueness of the definition of fractional operators and the influence of this problem on the properties of solutions of optimal-control problems for fractional systems. We consider two optimal-control problems for systems with lumped parameters whose dynamics is described by linear fractional integro-differential equations: the problem of search for a control with minimal norm at given control time and the time-optimal problem at given restriction on the norm of the control. We analyze the problems with the fractional derivative operators in the sense of Erdélyi–Kober, Katugampola, Atangana–Baleanu, and Caputo–Fabrizio. These problems are reduced to the $l$-moment problem and the well-posedness and solvability conditions are obtained. Several examples of the explicit calculation of optimal controls are given and the properties of the solutions obtained that depend on the type of the fractional derivative operator are compared.