توصيفگر ها :
انتگرال‌هاي كسري , مشتقات كسري , روش ريتز , مسائل حساب تغييرات كسري , مسائل كنترل بهينه كسري
چكيده فارسي :
حل مسائل حساب تغييرات كسري و كنترل بهينه كسري با روش غيرمستقيم، مبتني بر شرايط لازم بهينگي است. اين روش باعث ايجاد معادلات اويلر-لاگرانژ كسري مي‌شود. معادله اويلر-لاگرانژ كسري، يك معادله ديفرانسيل كسري از مرتبه α ، 0<α<1، در دو نقطه مجزا است كه حل آن به‌جز موارد ساده، بسيار پيچيده و در غالب موارد امكان‌پذير نيست. براي رفع اين مشكل، در اين پايان نامه، حل مسائل حساب تغييرات كسري و كنترل بهينه كسري با روش مستقيم ريتز ، مبتني بر سري‌هاي تواني كسري است. اين روش، مسأله حساب تغييرات كسري وكنترل بهينه كسري مورد‌نظر را به يك مسأله بهينه‌سازي پارامتري تبديل مي‌كند. خاطر نشان مي‌شود كه حل مسأله بهينه‌سازي پارامتري به‌دست آمده بسيار ساده‌تر از حل مسأله اصلي است. براي نشان دادن دقت، كارايي وكاربرد روش ارائه شده، مثال‌هاي گوناگوني را مورد بررسي و مطالعه قرار مي‌دهيم.
چكيده انگليسي :
In this thesis, we present a simple accurate numerical method for solving a class of fractional problems in the calculus of variations and fractional optimal control problems. Many real-life phenomena and practical systems can be modeled by various types of fractional-order differential equations. Typical examples are aerospace engineering, transmission lines, chemical processes, climate models, population dynamics, robotics, nuclear, economics, communication networks, biological models, manufacturing processes and power systems. In view of its history, fractional (noninteger order) calculus is as old as the classical calculus. Roughly speaking, there is just one definition of fractional integral operator, which is the Riemann–Liouville fractional integral. There are, however, several definitions of fractional differentiation, e.g., Caputo, Riemann–Liouville, Hadamard, And Wily differentiation. Each type of fractional derivative has its own properties, which make richer the area of fractional calculus and enlarge its range of applicability. There are two recent research areas where fractional operators have a particularly important role: the fractional calculus of variations and the fractional theory of optimal control. A fractional variational problem is a dynamic optimization problem, in which the objective functional, as well as its constraints, depends on derivatives or integrals of fractional order, e.g., Caputo, Riemann–Liouville or Hadamard fractional operators. This is a generalization of the classical theory, where derivatives and integrals can only appear in integer orders. If at least one noninteger (fractional) term exists in its formulation, then the problem is said to be a fractional variational problem or a fractional optimal control problem. The theory of the fractional calculus of variations was introduced by Riewe in 1996, to deal with nonconservative systems in mechanics. This subject has many applications in physics and engineering and provides more accurate models of physical phenomena. There are two main approaches to solve problems of the fractional calculus of variations or optimal control problems: indirect methods and direct methods. One involves solving fractional Euler–Lagrange equations or fractional Pontryagin-type conditions, which is the indirect approach; the other involves addressing directly the problem, without involving necessary optimality conditions, which is the direct approach. The emphasis in the literature has been put on indirect methods. Furthermore, Almeida et al. developed direct numerical methods based on the idea of writing the fractional operators in power series and then approximating the fractional problems with classical ones. In this thesis, we use a different approach to solve fractional variational problems and fractional optimal control problems, based on Ritz’s direct method. The idea is to restrict admissible functions to linear combinations of a set of known basis functions. We choose basis functions in such a way that the approximated function satisfies the given boundary conditions. Using the approximated function and its derivatives whenever needed, we transform the functional into a multivariate function of unknown coefficients. Recently, the Rayleigh–Ritz method was used based on Jacobi polynomials, to solve fractional optimal control problems. Several illustrative examples show that our results are more accurate and more useful than the ones introduced in literature. The simulation results demonstrate the effectiveness of the proposed method.
