توصيفگر ها :
مسئله كنترل بهينه كسري تأخيري , موجك برنولي , مشتق كسري كاپوتو , روش عددي
چكيده فارسي :
در اين پايان‌نامه، يك روش عددي كارا براي حل مسائل كنترل بهينه كسري تأخيري ارائه شده است. روش پيشنهادي، يك روش مستقيم مبتني بر موجك‌هاي برنولي است. ابتدا خواص چندجمله‌اي‌هاي برنولي و
روش ساخت موجك‌هاي برنولي را بيان مي‌كنيم. سپس با استفاده از ماتريس‌هاي عملياتي انتگرال كسري و تأخير متناظر با موجك‌هاي برنولي، مسئله كنترل بهينه كسري تأخيري مورد مطالعه را به يك مسئله بهينه سازي پارامتري تبديل مي‌كنيم. در نهايت با ارائه مثال‌هاي متعدد، عملكرد و دقت روش ارائه شده را مورد ارزيابي قرار مي‌دهيم.
چكيده انگليسي :
This thesis introduces a novel numerical method for addressing delay fractional optimal control problems (DFOCPs), characterized by a quadratic performance index. Optimal control problems (OCPs), play a crucial role across diverse fields including engineering, science, geometry, and applied mathematics, aiming to optimize performance criteria over permissible control and state functions while adhering to dynamic and state-control constraints. A specific subset, fractional optimal control problems (FOCPs), incorporates fractional derivative operators within the performance index or governing differential equations, essential for capturing complex dynamics inadequately modeled by integer-order derivatives. Fractional order dynamics have many applications in various fields of science and engineering such as viscoelastic materials, economics, statistical mechanics, bioengineering, and nanoparticle-substrate interfaces. Consequently, the development of robust analytical and numerical methods for solving fractional differential equations has garnered considerable attention. The thesis focuses on DFOCPs, where system dynamics are governed by delay fractional differential equations applicable to fields such as engineering, economics, power systems, and biology. Despite their broad relevance, research into delay fractional optimal control remains limited, yet crucial for practical applications in engineering and physics. Moreover, many DFOCPs lack exact analytical solutions, necessitating the exploration of accurate numerical methods. There are two main approaches to solving problems of the fractional calculus of variations or optimal control problems: indirect methods and direct methods. The indirect approach involves solving fractional Euler-Lagrange equations or fractional Pontryagintype
conditions, while the direct approach addresses the problem directly without involving necessary optimality conditions. The emphasis in the literature has been put on solving DFOCPs with Bernoulli wavelets which is considered a direct method. Initially, the problem is transformed into an equivalent form without delay, facilitating the application of an operational matrix for Riemann–Liouville fractional integration based on Bernoulli wavelets. This transformation reduces the problem to a set of algebraic equations, solved using Gauss–Legendre integration and Newton’s iterative method. In recent years, the construction and application of different wavelets such as the Chebyshev wavelet, second-kind Chebyshev wavelet, CAS wavelet, Legendre wavelet, and Bernoulli wavelet are a powerful mathematical tool for the discretization of selected problems. The main advantage of using wavelets is that the coefficient matrix of algebraic equations is sparse. other advantages of this method include post-discretization, computational efficiency making it suitable for high-dimensional problems, multiresolution properties enabling detailed solution capture across scales, and robust convergence even with substantial step sizes. The main advantage of our scheme is that with the use of only a few numbers of the Bernoulli wavelet basis elements, we obtain the approximation of the state vector and the control vector more accurately than some existing methods. Moreover, the operational matrix of fractional integration has large numbers of zero elements and it is sparse; thus, approximation with the Bernoulli wavelets has a short CPU time. In conclusion, this thesis contributes a novel computational framework employing Bernoulli wavelets to solve DFOCPs efficiently and accurately. The efficacy of the proposed approach is demonstrated through illustrative examples, showcasing its practical applicability and performance compared to existing methods. The simulation results confirm that the proposed numerical scheme based on the Bernoulli wavelets is efficient.
