Fractional calculus, which introduces derivatives and integrals of fractional order, has been demonstrated as an efficient modeling tool in a large number of engineering applications; mostly due to the non-local property and the memory effect of fractional operators. In the general case, however, most of the large-scale dynamic systems will include a set of differential equations that involves both fractional as well as ordinary differential equations. This paper is concerned with the modeling, simulation and optimization of ordinary-fractional dynamic systems. As our initial contribution, two dynamic systems are used to illustrate our modeling approach. The first case corresponds to a multi-compartmental fractional model representing the kinetics of a drug transfer within the organism (pharmacokinetics; PK). A formal fractionalization approach produces a consistent system (mass balance is enforced) and allows different fractional orders in the same system. A similar approach is used for our second illustrative example, where a dynamic model used for optimal dosage in diabetes treatment is reformulated as a fractional model. In both cases, the differential equations include ordinary and fractional operators, so that they both can be posed as a fractional-ordinary set of differential equations. For the simulation and optimization of those kinds of systems, most of the existing literature proposes a solution approach based on a numerical inverse Laplace transform algorithm. In this paper, however, we first introduce a generalized version of the predictor-corrector integration method, which can integrate simultaneously both fractional and ordinary differential equation in an efficient manner. This numerical integration tool is then used within an analytical/numerical optimization strategy. Such strategy combines the generalized optimality conditions for a fractional-ordinary system derived in this work (which represent a general version of the Euler-Lagrange Conditions for Optimality) and the gradient method to obtain optimal control profiles for the case-studies. To validate our analytical/numerical method, when possible, our numerical results are compared against the values reported in the literature. The optimal profiles are also used to show the effect of the fractional orders in the results.
Keywords: Multi-compartmental models, fractional calculus, optimal control, pharmacokinetics