Modeling, Simulation and Optimization of Combined Fractional-Ordinary Dynamic Systems

Abstract This work proposes an approach to the modeling and optimization of systems involving a coupled set of fractional and ordinary differential equations. We first present a generalized version of the predictor-corrector integration method, which can integrate simultaneously both fractional and ordinary differential equations. Further, we describe an analytical/numerical dynamic optimization strategy that combines the generalized optimality conditions for a fractional-ordinary system derived in this work, the generalized integration technique and the gradient method. The approach is illustrated through a compartmental model in pharmacokinetics as well as a fractional model for a thermal hydrolysis. In both cases, after we apply a formal fractionalization strategy, we propose a reformulation of the models to obtain fractional-ordinary dynamic systems. The systems obtained are further posed within an optimization framework and solved through our approach as fractional-ordinary optimal control problems. Our results show the theoretical and numerical consistency of our approach.