A Central Difference Numerical Scheme for Fractional Optimal Control Problems (0811.4368v1)

Abstract: This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a fractional derivative (FDs) at a time node point is approximated using a modified Gr\"{u}nwald-Letnikov approach. For the first order derivative, the proposed modified Gr\"{u}nwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.

Overview of a Numerical Scheme for Fractional Optimal Control Problems

The paper by Dumitru Baleanu, Ozlem Defterli, and Om P. Agrawal introduces a modified numerical scheme for solving Fractional Optimal Control Problems (FOCPs), emphasizing the use of fractional derivatives defined in the Riemann-Liouville sense. Building on Agrawal's formulation from 2004, the authors propose an improved approach to fractional dynamics in optimal control. The study provides both a theoretical foundation and numerical experimentation to underscore the potential of the proposed method.

The research specifically focuses on a class of FOCPs where a modified Gr{\"u}nwald-Letnikov scheme offers a central difference approximation for fractional derivatives. The scheme divides the entire time domain into sub-domains, applying the central difference approximation at each time node. This results in a set of algebraic equations that are efficiently solvable using direct numerical techniques.

Two key numerical examples are analyzed in the study: one that is time-invariant and another that varies with time. The results present compelling evidence that as the order of the fractional derivative ($\alpha$) nears an integer value, the scheme effectively bridges the gap with traditional integer-order systems. Moreover, the solutions demonstrate convergence when sub-domain sizes are decreased. This convergence hints at the potential for this technique to contribute to the stabilization of numerical methods for fractional differential equations and optimal control problems.

Key Insights and Mathematical Formulation

The study initially revisits the concept of Fractional Dynamic Systems (FDS), where the dynamics are governed by Fractional Differential Equations (FDEs). The optimal control problems tackled involve minimizing a performance index subject to dynamic constraints that incorporate fractional derivatives.

The paper eloquently defines the Hamiltonian formulation for FOCPs, positing the Hamiltonian as: $H(x, u, \lambda, t) = f(x,u,t) + \lambda g(x,u,t)$ Subsequent necessary conditions for control optimization are derived, with systems expressed in fractional notation: $$_0D_t^\alpha x = \frac{\partial H}{\partial \lambda}$$

For the numerical scheme's effectiveness, the authors introduce central difference approximations specific to fractional derivatives, defined recursively for coefficients: $$\omega_j^{(\alpha)} = \left( 1 - \frac{\alpha+1}{j} \right) \omega_{j-1}^{(\alpha)}$$

Numerical Experiments and Convergence

Demonstrating the proposed method's capacity, the researchers solve time-invariant and time-variant FOCPs. The convergence of these solutions, notably slower at lower $\alpha$ values, underscores the stability promise while highlighting areas of slow convergence that warrant further investigation.

Specifically, the time-invariant problem examines a quadratic performance index: $J(u) = \frac{1}{2} \int_0^1 [ x^2(t) + u^2(t) ] dt$ subject to dynamic constraints formulated in fractional terms.

Through these experiments, as $\alpha$ approaches 1, results confirm that the presented numerical solutions align with those expected from integer-order systems, thus verifying the scheme's validity in both fractional and integer contexts.

Implications and Future Directions

This paper presents a meticulous scheme with significant implications for fractional calculus, particularly in systems represented by non-integer dynamics. The convergence towards solutions for integer order systems suggests practical applications in fields like engineering, where the accuracy of dynamic system modeling is paramount. While the slow convergence indicates room for further optimization of the numerical method, the current framework offers a foundation for the application of fractional calculus in broader disciplines.

The promise of this numerical scheme could steer future research into enhancing the computational efficiency and exploring its application across complex systems, potentially catalyzing developments in fractional dynamics modeling in technological and scientific applications.