Fractional Optimal Control of Distributed Systems

This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a Fractional Optimal Control Problem (FOCP) is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). The scheme presented rely on reducing the equations for distributed system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. This involves discretizing the space domain into several segments, and writing the spatial derivatives in terms of variables at space node points. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] for scalar case is extended for the vector case. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is also descretized into several segments. For the linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for various order of fractional derivatives and various order of space and time discretizations. Numerical results show that for the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.