New pure multi-order fractional optimal control problems with constraints: QP and LP methods

Novel methods for finding the optimal controls of new types of fractional optimal control problems with Riemann–Liouville performance indices and systems comprised of subsystems with Caputo derivatives are introduced. Pure fractional quadratic optimal control problems are modeled as quadratic programming (QP) by using a new idea and a state-control parameterization method. After formulating each linear or nonlinear type, its QP model is derived by which the QP solver in MATLAB can be used to obtain the solutions. There is no need for such operations as defining costate variables, deriving optimality conditions, etc. New concepts such as fractional boundary constraints and Riemann–Liouville isoperimetric constraints, are introduced. Multiple problems in different scenarios are investigated and numerous graphs and numerical results are presented. Pure fractional linear control problems with Riemann–Liouville performance indices and fractional systems are modeled as linear programming (LP) without discretization for the first time. Using the LP solver in MATLAB, the optimal solutions of the fractional/integer linear control problems such as bang–bang (or On–Off) and minimum fuel optimal control systems are obtained. Fractional types of the real-world problems such as container cranes and drug scheduling of cancer chemotherapy, are studied.