INTRODUCTION 🔍
Fractal–fractional derivatives have emerged as powerful tools in mathematical modeling, allowing for a nuanced description of memory and hereditary properties in complex dynamical systems. This paper focuses on the development of a novel numerical computation technique for solving fractal–fractional optimal control problems incorporating Caputo–Fabrizio derivatives. By leveraging this advanced fractional calculus, the research addresses highly nonlinear systems where classical methods fall short. These problems often include equality and inequality constraints on the state variables, reflecting real-world limitations and resource boundaries. The approach taken in this work systematically transforms these infinite-dimensional control problems into finite-dimensional ones using the control parametrization technique. This transformation is pivotal in enabling numerical analysis and optimization, forming the foundation of the proposed solution strategy. The motivation stems from the need to solve real-life control problems with better accuracy and reduced computational complexity, especially in fields like epidemiology, where modeling the spread of diseases such as AIDS demands high-fidelity representations of memory effects.
FRACTAL–FRACTIONAL OPTIMAL CONTROL FORMULATION 🧩
At the heart of this study is a carefully defined class of fractal–fractional optimal control problems governed by Caputo–Fabrizio derivatives. These derivatives allow for a smooth, non-singular kernel formulation, capturing transient behaviors and long-term dependencies without the difficulties associated with singularities in traditional fractional models. The control problems are subjected to both equality and inequality constraints, mirroring real-world limitations such as bounded resources or mandated safety thresholds. The formulation ensures flexibility and generality, making it adaptable to a wide variety of scientific and engineering applications. By establishing a rigorous mathematical framework, the paper sets the stage for applying numerical techniques that maintain accuracy while managing computational demands. The elegance of the Caputo–Fabrizio derivative in modeling systems with memory, coupled with the generality of the control constraints, ensures that the approach is not only theoretically sound but practically significant.
CONTROL PARAMETRIZATION APPROACH 🎯
A significant contribution of this research lies in the application of the control parametrization technique to convert complex infinite-dimensional fractal–fractional optimal control problems into manageable finite-dimensional ones. By discretizing the control space, the problem becomes a sequence of standard optimization problems with decision variables represented as control parameters. This technique facilitates practical computation and algorithm development while preserving the critical dynamics of the original system. It allows for iterative refinement, making it suitable for gradient-based optimization procedures. Importantly, control parametrization serves as a bridge between the abstract mathematical formulation and concrete numerical computation, enabling the real-time application of optimal control in dynamic environments such as epidemic spread, industrial systems, and energy networks.
GRADIENT COMPUTATION AND AUXILIARY SYSTEMS ⚙️
To support the optimization framework, the gradients of the cost and constraint functions with respect to the decision variables are analytically derived. This derivation is crucial for the convergence and efficiency of gradient-based optimization algorithms. The gradients are obtained by solving specially constructed auxiliary fractal–fractional systems that mirror the behavior of the original control problem. These auxiliary systems are carefully modeled using the properties of the Caputo–Fabrizio derivatives, ensuring that the gradients are both accurate and computationally feasible. The ability to compute these gradients efficiently is vital for scaling the algorithm to higher-dimensional problems and maintaining robustness across different applications.
NUMERICAL SCHEME FOR SOLVING FRACTAL–FRACTIONAL SYSTEMS 🧮
A third-order numerical scheme is developed and employed to solve the fractal–fractional systems involved in the control problems. This high-order method ensures enhanced accuracy and stability, which is particularly important when dealing with systems governed by memory-dependent dynamics. The scheme integrates the nuances of the Caputo–Fabrizio derivative, including its exponential kernel and non-local effects. This ensures that the numerical approximation closely follows the behavior of the underlying system. By validating the scheme across multiple examples, the study confirms that it can efficiently handle a wide range of dynamic scenarios without significant loss of precision, thus providing a reliable backbone for the overall computational framework.
APPLICATION TO EPIDEMIC CONTROL MODELS 🧬
One of the compelling demonstrations of the developed technique is its application to an optimal control problem involving the spread of acquired immunodeficiency syndrome (AIDS). By modeling the epidemic using fractal–fractional derivatives, the system effectively captures the disease’s progression over time, accounting for population behavior and treatment memory. The control parametrization method, combined with the third-order numerical scheme and gradient-based optimization, successfully minimizes the cost function while adhering to public health constraints. This real-world application highlights the framework’s potential to influence policy and healthcare strategies, providing quantitative tools for controlling diseases with complex transmission dynamics.
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HASHTAGS
#FractalFractionalCalculus, #OptimalControl, #CaputoFabrizioDerivative, #FractionalDifferentialEquations, #NumericalOptimization, #ControlParametrization, #GradientBasedMethods, #AIDSEpidemicModel, #FractionalModeling, #ThirdOrderScheme, #ScientificComputing, #AppliedMathematics, #ComplexSystems, #FractionalControlTheory, #MemoryEffects, #ComputationalMathematics, #MathematicalModeling, #EngineeringOptimization, #NumericalSimulation, #FractionalEpidemiology,
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