(245f) Optimization of Dynamic Systems Including Ordinary and Fractional Differential Equations

Fractional calculus is a generalization of ordinary calculus which introduces derivatives and integrals of fractional order. Several authors have recently shown that fractional calculus is a powerful modeling tool to represent the behavior of a number of mechanical and electrical dynamic systems (Magin, 2006; Sabatier et al, 2007). Several works also describe and/or study the non-locality property and the memory effect of fractional calculus operators (Magin, 2006; Herrmann, 2011; Sun et al, 2011; Constantinescu and Soicescu, 2011; Du et al, 2013). Our approach considers, though, that 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 work is concerned with both the formulation and the numerical solution strategies of optimal control problems including fractional and ordinary differential equations. First, we focus on the use of fractional calculus as a mathematical tool with potential applications to chemical engineering; in particular, to the area of chemical reaction engineering. To illustrate this idea, we consider a reactive system which exhibits anomalous kinetics. Anticipating a potential memory effect on the dynamics of some of the state variables of such system, we show that it can be represented by a set of differential equations which includes both fractional and ordinary differential equations. To obtain the model parameters and the orders of the fractional differential equations, a non-liner fitting approach is coupled to a numerical integration technique.

Then, given the combined set of differential equations representing the behavior of the system, the corresponding optimal control problem is formulated. We describe theoretical and numerical suitable techniques for solving it. An analytical/numerical strategy that combines the optimality conditions of the problem and the gradient method as well as the predictor-corrector integrator is used to obtain optimal control profiles for the case-study. The optimal profiles show the performance of the analytical/numerical solution approaches proposed and the effect of the orders of the differential equations in the optimal results.

References

Constantinescu, D., M. Stoicescu, 2011, Fractal dynamics as long range memory modeling technique, Physics AUC, 21: 114-120

Du, M., Z. Wang, H. Hu, 2013, Measuring memory with the order of fractional derivative, Scientific Reports, 3:3431.

Herrmann, R., 2011, Fractional Calculus. An introduction for physicists, World Scientific Publishing Co., Hackensack, NJ, USA.

Magin, R., 2006, Fractional calculus in bioengineering, Begell House Inc. Publishers, Redding, Connecticut, USA.

Sabatier, J., O. P. Agrawal, J. A. Teneiro-Machado, 2007, Advances in fractional calculus. Theoretical developments and applications in Physics and engineering, Springer, Berlin, Germany.

Sun, H. G., W. Chen, H. Wei, Y. Q. Chen, 2011, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, The European Physical Journal Special Topics, 193:185â??192.